MRA(from subdivision viewpoint)

Jyun-Ming Chen

Spring 2001

Nested Spaces

• : record the combined effect of splitting and averaging done to the initial control points to achieve the limit f(x)

• The same limit curve can be defined from each iteration

• Using matrix notation

We will show that every subdivision scheme gives rise to refinable scaling functions and, hence, to nested spaces

: some yet undetermined functions;(later we’ll show they are the scaling functions)

Nested Spaces (cont)• Subdivision (refinement)

matrix Pj

– Represent the combined effect of splitting and averaging (both are linear operations)

• refinement relation for scaling functions– Observe the similar relation

for Haar and Daub4

Nested Space (cont)

• The refinement relation states that each of the coarser scaling functions can be written as a linear combination of the finer scaling functions.

• This linear combination depends on the subdivision(refinement) scheme used.

Nested Space

• Define space Vj that includes all linear combinations of scaling functions (of j), whose dimension denoted by v(j)

• Then

• From

Pj is a v(j) by v(j-1) matrix

jjv

jjj xxV 1)(10 ,),(),(span

jj VV 1

Wavelet Space

• Define wavelet space Wj to be the complement of Vj in Vj+1 ; implying– Any function in Vj+1 can be written as the sum of a unique function i

n Vj and a unique function in Wj

– The dimensions of these spaces are related

• The basis for Wj are called wavelets

• The corresponding scaling function space

• Also write wavelet space Wj as linear combination using basis of next space Vj+1

• From before,

Wavelet Space (cont)

• Combining, this is called the two-scale relation.

Splitting of MRA Subspaces

VN

VN-1 WN-1

VN-2 WN-2

VN-3 WN-3

Example: Haar

2

1

2

12

1

2

1

210

210

021

021

2121

2121

2121

2121

210210

210210

021021

021021

1000

0100

0010

0001

210210

210210

021021

021021

1)dim(,1)dim(

2)dim(,2)dim(

4)dim(

00

11

2

WV

WV

V

224125379

5

3

7

9

2c

2

2

24

28

1

1

d

c

4

12

0

0

d

c

22211 QP

11100 QP

Two-Scale Relations (graphically)

2

130

2

131

20

2

130

2

131

20

Analysis Filters

• Consider the approxi-mation of a function in some subspace Vj

• Assume the function is described in some scaling function basis

• Write these coefficients as a column matrix

• Suppose we wish to create a lower resolution version with a smaller number of coefficients v(j-1), this can be done by

Aj is a constant matrix of dimension v(j-1) by v(j)Tj

jvjj ccc ][ 1)(0

jjcxf )(

Analysis/Decomposition

• To capture the lost details as another column matrix dj-1

Bj is a constant matrix of dimension w(j-1) by v(j), relating to Aj

• This process is called analysis or decomposition

• The pair of matrices Aj and Bj are called analysis filters

)1()()1( jvjvjw

Synthesis/Reconstruction

• If analysis matrices are chosen appropriately, original signal can be recovered using subdivision matrices

• This process is called synthesis or reconstruction

• The pair of matrices Pj and Qj are called synthesis filters

Closer Look at Synthesis

• Performing the splitting and averaging to bring cj-1 to a finer scale

• A perturbation by interpolating the wavelets

Filter Bank

• Doing the aforementioned task repeatedly

• Recall Haar (next page)

1100

0011

2

12A

1100

0011

2

12B

Haar (Analysis)

21

21

2

212100

002121

2

2

212100

002121

24

28

]5379[

cd

cc

c T

A2

B2

Haar (Synthesis)

2

2

210

210

021

021

24

28

210

210

021

021

5

3

7

9

P2 Q2

2222 ,

:Note

QBPATT

Relation Between Analysis and Synthesis Filters

• In general, analysis filters are not necessarily transposed multiples of the synthesis filters (as in the Haar case)

1111)( jjjjjj dccxfjjjjjj cBcA 11

jjjjj BA 11

Analysis & Synthesis Filters

• Dimension of filter matrices– Aj: v(j-1) by v(j);– Bj: w(j-1) by v(j)– Pj: v(j) by v(j-1)– Qj: v(j) by w(j-1)

• Hence are both square

• … and should be invertible

• Combining:

• We get:

Orthogonal Wavelets

Orthogonal Wavelets

• Scaling function orthogonal to one another in the same level• Wavelets orthogonal to one another in the same level and in

all scales• Each wavelet orthogonal to every coarser scaling function • Haar and Daubechies are both orthogonal wavelets

Implication of Orthogonality

• Two row matrices of functions

• Define matrix

• Has these properties:

Implication of Orthogonality

Changing subscript to j-1

Orthonormal …

• Scaling functions– Scaling functions are o

rthonormal only w.r.t. translations in a given scale

– Not w.r.t. the scale (because of the nested nature of MRA)

• Wavelets– The wavelets are ortho

normal w.r.t. scale as well as w.r.t. translation in a given scale