Lifting

Part 2: Subdivision

Ref: SIGGRAPH96

Subdivision Methods

On constructing more powerful predictors …

Subdivision methods

• Often referred to as the cascade algorithm• Systematic ways to build predictors

– Concentrate on the P box

• Types:– Interpolating subdivision– Average-interpolating– B-spline (more later)– …

Interpolating Subdivision

First proposed by Deslauriers-Dubic

Basic Ideas

• In general, use N (=2D) samples to build a polynomial of degree N-1 that interpolates the samples

• Calculate the coefficient on the next finer level as the value of this polynomial– e.g., Lagrange polynomial (or Neville’s algorithm)

• Order of the subdivision scheme is N• This can be extended to accommodate bounded

interval and irregular sampling settings.

Math Review: Lagrange Polynomial

• The unique n-th degree polynomial that passes through (n+1) points can be expressed as follows:

)()(

)()(

0 0i

n

i ji

jn

ijj

xfxx

xxxP

niii xfx ,...,0))(,(

Linear and Cubic Interpolation

Order = 2 Order = 4

2

1

2

1

16

116

9

16

9

16

1

Numerical Example: Cubic Interpolation Stencil

48104… …

48104 877… …

9/16 9/16 -1/16-1/16

Scaling Functions

• All scaling functions at different levels are translates and dilates of one fixed function:– the fundamental solution (so named by the origi

nal inventor, Deslauriers-Dubuc) of the subdivision scheme

• Obtained by cascade algorithm

)(x

Cascading (linear interpolation)

3,23,22,22,21,21,20,20,2 dsdsdsds

00000100

000010 21

21

0,1,0,0

0,0,0,0 .5, .5, 0,0

0, .5, 1, .5, 0,0,0,0

1,11,10,10,1 dsds

0100

21

21 10

41

21

43

43

21

41 10

0,1

0,0 .5, .5

0, .5,1, .5

.25, .75, .75, .25

0, .25, .5, .75, 1, .75, .5, .25

Cascading !

Compare with what we said before …

• From forward transform– Hi-wire: coarsened signal

– Lo-wire: difference signal

• Subdivision: Inverse transform with zero detail• Cascading: apply delta sequence to get impulse

response (literally)– Hi-wire: scaling functions

– Lo-wire: wavelets

Interpolating Scaling Functions

Properties of Scaling Functions

• Compact support– [-N+1, N-1]

• Interpolating

• Smoothness– N large, smoother …

• Polynomial reproduction– Polynomials up to

degree N-1 can be expressed as linear combinations of scaling functions

)(x

0for 0)(

1)0(

kk

Properties of Scaling Functions

• Refinability

)(x

1

)1(

)2()(N

Nll lxhx

zero-non are terms12Only

result thecall;

fromn subdivisio of step one do

:follows as obtained

,10,,0

N

shs

h

llkk

l

16

1,0,

16

9,1,

16

9,0,

16

1 :cubic

2

1,1,

2

1 :linear

l

l

h

h

ransform wavelet tinverse in thefilter

pass low theof FIR thedescribes lh

Computing the filter coefficients

0010

3,02,01,00,0 ssss

000010 21

21

1,0,12

121 ,1,

llh

0100

161

169

169

161 0100

3,,316

1169

169

161 ,0,,1,,0,

llh

N=2

N=4

Refinement Relations

kjk

kllj

ljl

klkj

shs

xhx

,2,1

,12, )()(

kjk

kllj

ljl

klkj

shs

xhx

,2,1

,12, )()(

21

21 1

0010sj

00000100upsampling

000010 21

21sj+1

21

21 1

21

21 1

21

21 1

21

21 1

1321sj

01030201upsampling

21

25

23 12321sj+1

21

21 1

21

21 1

21

21 1

21

21 1

21

21 1

21

21 1

21

21 1

Average-Interpolating Subdivision

Proposed by Donoho (1993)

Basic Ideas

Think of the signals as the intensity

obtained from CCD

Meaning of Signal sj,k

area = Sj,k (width)

Sj,k : the average signal in this interval

p(x)

CCD sensor

Averaging-interpolating subdivision (constant)

• Which (constant) polynomial would have produced these average?

• Subdivide according to the (implied) constant polynomial

Order = 1

Average-interpolating subdivision (quadratic)

Order = 3

defines the (implied) quadratic curve

produce the finer averages accordingly

Average-Interpolating (N=3)

j

j

j

j

j

j

k

kkj

k

kkj

k

kkj

dxxps

dxxps

dxxps

2)2(

2)1(1,

2)1(

2,

2

2)1(1,

)(

)(

)(

j

j

j

j

k

kkj

k

kkj

dxxps

dxxps

2)1(

2)(12,1

2)(

22,1

21

21

)(2

)(2

jk 2)1( jk 2 jk 2)1( jk 2)2(

jk 2)( 21

p(x) is the (implied) quadratic

polynomial

p(x) is the (implied) quadratic

polynomial

The coefficient “2” is due to half width

Average-Interpolating (N=3)

x

k jdyypxP

2)1()()( 3rd degree polynomial

)2)2((

)2)1((

)2(

)2)1((0

1,,1,

,1,

1,

jkjkjkj

jkjkj

jkj

j

kPsss

kPss

kPs

kP

)2)(()2)1((2

)2()2)((2

21

12,1

21

2,1

jjkj

jjkj

kPkPs

kPkPs

Define

4 conditions: P(x) can be d

etermined

Numeric Example (N=3)

)3(12

)2(8

)1(3

)0(0

1,,1,

,1,

1,

Psss

Pss

Ps

P

kjkjkj

kjkj

kj

125.5)4375.58(2

)5.1()2(2

875.4)34375.5(2

)1()5.1(2

12,1

2,1

PPs

PPs

kj

kj

Solve for P(1.5) =5.4375 using Lagrange polynomial (next page)

…

0 1 2 31.5

…

4531,,1, kjkjkj sss

4.875 5.125

Lagrange Polynomial

)3()23)(13)(03(

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

)32)(12)(02(

)3)(1)(0(

)1()31)(21)(01(

)3)(2)(0()0(

)30)(20)(10(

)3)(2)(1()(

Pxxx

Pxxx

Pxxx

Pxxx

xP

5.4375

12)23)(13)(03(

)25.1)(15.1)(05.1(8

)32)(12)(02(

)35.1)(15.1)(05.1(

3)31)(21)(01(

)35.1)(25.1)(05.1(0

)30)(20)(10(

)3)(2)(1()5.1(

xxx

P

Details

Derive Weighting (N=3)

)3(

)2(

)1(

)0(0

1,,1,

,1,

1,

Psss

Pss

Ps

P

kjkjkj

kjkj

kj

?)5.1( Solve P

)3()23)(13)(03(

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

)32)(12)(02(

)3)(1)(0(

)1()31)(21)(01(

)3)(2)(0()0(

)30)(20)(10(

)3)(2)(1()(

Pxxx

Pxxx

Pxxx

Pxxx

xP

1,161

,168

1,1617

1,,1,161

,1,169

1,169

1,,1,

,1,1,

)()(

)()23)(13)(03(

)25.1)(15.1)(05.1(

)()32)(12)(02(

)35.1)(15.1)(05.1()(

)31)(21)(01(

)35.1)(25.1)(05.1()5.1(

kjkjkj

kjkjkjkjkjkj

kjkjkj

kjkjkj

sss

ssssss

sss

sssP Check:

If sj,k-1 = sj, k= sj,k+1 = x,

P(1.5) = 1.5x = 24x/16

Consider in-place Computation

125.5)4375.5(2

)5.1()2(2

875.4)4375.5(2

)1()5.1(2

,1,

12,1

1,

2,1

kjkj

kj

kj

kj

ss

PPs

s

PPs

12,12,1

1,,1,

kjkj

kjkjkj

ss

sss

•Solution 1 : compute sj+1,2k+1 first

•Not a good solution… dependent on execution sequence

Problem: occupy the same piece of memory

Observe that …

)2)((222

)2)((2)2()2)1((2

:difference

)2()2)1((2

:average

21

,1,

21

2,112,1,

,2,112,1

jkjkj

jjj

kjkjkj

kjjjkjkj

kPss

kPkPkP

ssd

skPkPss

Utilize inverse Haar transform !

1,4

11,4

1

1,161

,168

1,1617

,1,

2,112,1,

,2,112,1

222

:difference2

:average

kjkj

kjkjkjkjkj

kjkjkj

kjkjkj

ss

sssss

ssd

sss

1,41

1,41

,AI )( kjkjkj sssP 1,41

1,41

,AI )( kjkjkj sssP

Closed form of quadratic PAI

Three-Stage Lifting

25.043

)(

52/)(

41

1,1,41

2,112,1,

2,112,1,

kjkjkjkjkj

kjkjkj

ssssd

sss

kjs 2,1

12,1 kjs

kjs ,

kjd ,

125.5875.425.0)(

875.42

25.05)(

2,1,12,1

,,2,1

kjHaarkjkj

kjHaarkjkj

sPds

dUss

Numerical Example (N=3)

3, 5, 4, 3

0, 0, 0, 0

0.5, 0.25, –0.5, –0.25

2.75, 4.875, 4.25, 3.125

3.25, 5.125, 3.75, 2.875

Merged Result: 2.75, 3.25, 4.875, 5.125, 4.25, 3.75, 3.125, 2.875

AI Subdivision

AI Scaling Function by Cascading (N=3)

0, 1, 0, 0

0, 0, 0, 0

0.25, 0, –0.25, 0

-0.125, 1, 0.125, 0

0.125, 1, -0.125, 0

Merged Result: -0.125, 0.125, 1, 1, 0.125, -0.125, 0, 0

Remark

• Recall inverse Haar preserves average …

• Implying …

• More about this later

Properties of Scaling Functions

• Compact support– [-N+1, N]

• Average-interpolating

• Polynomial reproduction– Up to degree N-1

• Smoothness:– continuous of order R(N)

• Refinability:– Obtained similarly as i

n interpolating subdivision

1283

1283

6411

6411

6411

6411

1283

1283

81

81

81

81

,,,,1,,,,

(1,5) CDF :quartic

,,1,1,,

(1,3) CDF :quadratic

l

l

h

h

0,

1)( k

k

kdxx

kjk

kllj

ljl

klkj

shs

xhx

,2,1

,12, )()(

Average-interpolating scaling functions

Summary

• Types of Predictors:– Interpolating

– Average-interpolating

– B-spline

• So far, we only considered subdivision in inverse transform. How about its role in forward transform?

• Roles of Predictors– In inverse transform

• Subdivision

– In forward transform:• Predict results to

generate the difference signal (low-wire)

• More …– On constructing more

powerful P boxes– Define “power”!?

MRA and Lifting(part I)

MRA Properties

• Scaling functions at all levels are dilated and translated copies of a single function

1

1

,

,

)2( then )( if :Dilation

)2( then )( if :nTranslatio

:Nestedness

),(span

)2()(

jj

jj

j

jj

ljj

jlj

VxfVxf

VkxfVxf

VV

ZlxV

lxx

)(x

Order of an MRA

• The order of MRA is N if every polynomial of degree < N can be written exactly as a linear combination of scaling functions of a given level

• The order of MRA is the same as the order of the predictor used to build the scaling functions

Graphing by Cascading

• Scaling functions: delta sequence on hi-wire

• Wavelets: delta sequence on lo-wire More on this later

Homeworks

• Derive the weights for cubic interpolation

• Implement cascading to see scaling functions (and wavelets) at different levels

• Use lifting to process audio data– Provide routines for read/write/plot data– denoising radio recordings (WAV)

undecided

Convention:

• Smaller index, smaller data set (coarser)

• 2D lifting the same as classical?!

• Lifting and biorthogonality!?

From lifting-2

• Filter coefficient • Refinement relations follow from the fact that subdivision from level 0 with s0,k and level 1 with s1,k should be the same.