2D Wavelet

Digital Image Processing II

2D Wavelets for Different Sampling

Grids and the Lifting Scheme

Miroslav VrankićUniversity of Zagreb, Croatia

Presented by: Atanas Gotchev


Lecture Outline

1D wavelets and FWT2D separable wavelets2D nonseparable wavelets– different sampling grids

Lifting scheme– easy to construct filter banks


Two-Channel Filter Bank

2

2

2

2

x[n]H0

H1

G0

G1

x0[n]

x1[n] x[n]^

Analysis Synthesis

][][ˆ 0nnxnx −=

LP channel: H0 and G0HP channel: H1 and G1PR condition:


FWT: Analysis Filter Bank

Fast wavelet transform enables efficient computation of DWT coefs.Iteration of the analysis FB on the low-pass channelDWT coefficients are computed recursively!


FWT: Analysis Filter Bank


Synthesis Bank


Complexity of FWT

Number of operations proportional to:N – size of dataL – length of filters in the filterbank (scaling and wavelet vectors)


Separable wavelet transforms

products of 1D wavelet and scaling functionsϕ(x,y) = ϕ(x)ϕ(y)ψΗ(x,y) = ψ(x)ϕ(y)ψV(x,y) = ϕ(x)ψ(y)ψD(x,y) = ψ(x)ψ(y)


2D separable FWT


Example: Symlets wavelets

See functionssymaux,dbauxin WaveletToolbox


Wavelet and the Scaling Function


2D wavelets and scaling function


Sampling in 2D

Image is split into several groups of pixels (phases)Not as straightforward as in 1DMany ways to split an image– Separable– Quincunx– Hexagonal...


Quincunx Downsampling

n2

n1

Image is split into two phases (cosets)Simplest nonseparable sampling scheme


Subsampling Matrix

Basis vectors form the unit cellSubsampling matrix (dilation matrix) defines the sampling operation

1 11 1

⎡ ⎤= ⎢ ⎥−⎣ ⎦

D(1,-1)

(1,1)

n2

n1


Subsampling Matrix

Defines the sampling gridFor a 2D grid, D is a 2x2 matrix.

There are M = |det(D)| image phasesand also M samples in the unit cell.For the quincunx case, M = 2.– Quincunx PR FB needs M = 2 channels.


2D Subsampling Operation

D defines the sampling gridTake one coset of the imageRenumber it to fit on the integer grid

1 11 2 1 2

2 2( , ) ( , ), where D

k nx n n x k k

k n⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

D


Quincunx Subsampling Operation

For the quincunx case:

1 1 1 2

2 2 2 1

1 2 1 2 2 1

1 11 1

1 11 1

( , ) ( , )D

k n n nk n n n

x n n x n n n n

⎡ ⎤= ⎢ ⎥−⎣ ⎦

+⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ −−⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= + −

D


Downsampling is actually...

”reading” the image along the new axes.45° rotation for the quincunx case

(1,-1)

(1,1)

n2

n1 (1,0)

(0,1)

n2

n1


To take the second phase...

move the new axes by (1,0)...to the next element of the unit cell.

(1,0)

(0,1)

n2

n1

(2,-1)

(2,1)

n2

n1


Quincunx Polyphase Decomposition

Phase 2

Phase 1

Counterclockwise rotation


Separable Sampling

4 elements of the unit cellImage is split into 4 phasesRequires 4 channels

of the PR filter bank(2,0)

(0,2)

n2

n12 00 2⎡ ⎤

= ⎢ ⎥⎣ ⎦

D


Hexagonal Sampling

4 elements of the unit cellImage is split into 4 phasesRequires 4 channels of the PR filter bank

(1,-2)

(1,2)

n2

n1 1 12 2

⎡ ⎤= ⎢ ⎥−⎣ ⎦

D


Voronoi cell

Voronoi cell consists of points closer to the origin...than to any other point of the given lattice.Quincunx Voronoi cell n2

n11

1


Effects in the Frequency Domain

Downsampling is defined with a D matrix

To avoid aliasing...signal should be bandlimited to Voronoi cell of the lattice defined by 2πD-T

T

( )

1( ) ( ) ( ) ( 2 )det T

DN

X X X π⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

−

∈

= ↓ = −| | ∑k D

ω D ω D ω kD

1

2

ωω⎡ ⎤

= ⎢ ⎥⎣ ⎦

ωwhere


Bandlimiting

Properly bandlimited signal for quincunx downsampling

ω1π

πω2

ω2

ω1π 2π

π

2π


Quincunx downsampling

Input image has been properly bandlimited

Spectrum support of the downsampled image

ω2

ω1π 2π

π

2π

ω2

ω1π 2π

π

2π


Quincunx upsampling

(1,-1)

(1,1)

n2

n1(1,0)

(0,1)

n2

n1

1( ) if ( )( )0 otherwiseU

x LATx−⎧ ∈⎪= ⎨

⎪⎩

D n n Dn


Upsampling effect on Z-transform

)()()()()( 1 DDk

k

n

n

n

n

zzkznDznz XxxxX UU ==== −−−− ∑∑∑

212

1

212

1 nnnn

zzzz

=⎥⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡

nz

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡

2212

21112221

1211

21

21

2

1dd

dddddd

zzzz

zzDz

kDDk zz )(=Exercise: prove that


Frequency transformation

ωz je→

ωDDzTj

ddj

ddj

dd

dd

eee

zzzz

=⎥⎦

⎤⎢⎣

⎡→⎥

⎦

⎤⎢⎣

⎡=

+

+

)(

)(

21

21222112

221111

2212

2111

ωω

ωω

)()( ωDω TU XX =Conclusion:


Quincunx upsampling

( ) ( )TUX X=ω D ω( )X ω

ω2

ω1π 2π

π

2π

ω2

ω1π 2π

π

2π


Iterated quincunx upsampling

π

πω2

ω1

T( ) ( )UX X=ω D ω

π

πω2

ω1

( )2 T( ) ( )UX X=ω D ω

π

πω2

ω1

( )3 T( ) ( )UX X=ω D ω


The Lifting Scheme

Simple way to construct filter banksEasy to satisfy PR requirementComputationally efficient

X(z)

P(z)

D(z)

A(z)

X(z)

2

2

P(z)

+

2

2^-

U(z)

+

U(z)

-z-1z-1


The Lifting Scheme

Basic structure:– Polyphase decomposition– Predict stage (dual lifting step)– Update stage (primal lifting step)

X(z)

P(z)

D(z)

A(z)

X(z)

2

2

P(z)

+

2

2^-

U(z)

+

U(z)

-z-1z-1


Predict stage

Prediction of the second phase sample...based on a number of samples from the first phase.Wavelet coefficients are obtained as...a prediction error.

Smooth signal...gives small details.

X(z)

P(z)

D(z)

2

2-

z-1


Update stage

Input: detail coefs.Output is used to create approximation coefs.Average value of the input image must be retained. X(z)

P(z)

D(z)

A(z)2

2-

U(z)

+z-1


Lifting Scheme in 2-D

X(z1,z2)

P(z1,z2)

D

A

P(z1,z2)

+

D

D^-

U(z1,z2)

+

U(z1,z2)

-

z1z1-1

X(z1,z2)

Xe

Xo

D

D

similar structure as 1-D2D polyphase decomposition2D filters


Quincunx FB Example

Lifting scheme based on quincunx interpolating filtersJ. Kovačević & W. Sweldens: Wavelet Families of Increasing Order in Arbitrary Dimensions. IEEE Trans. Image Proc., vol. 9, no. 3, pages 480-496, March 2000.


Predict Filters

Neville interpolating filterssymmetric interpolation neighborhoods

example of a second order P filter:

n2

n1

n2

n1

n2

n1

n2

n1

n2

n1

n2

n1

n2

n1

12

11

12

11212 25.025.025.025.0),( −−−− +++= zzzzzzP

n2

n1


Supports of the Prediction Filters


Update Filters

updates the average value of the input image

based on the corresponding predict filter

*1 2 1 2

1( , ) ( , )2N NU z z P z z=


Transfer Functions for P4 and U2

Synthesis LP

Analysis LP Analysis HP

Synthesis HP


Wavelet and Scale for P4 and U2

Analysis wavelet

Synthesis scale

Analysis scale

Synthesis wavelet


Wavelet Decomposition Tree

AJ-1

DJ-1

AJ-2

DJ-2

AJ-3

DJ-3


Separable Versus Nonseparable

Nonseparable– higher complexity– more freedom in FB design– different directional properties

Separable– widely used– simple realization based on 1D filter banks


Quincunx Wavelets

Simplest nonseparable sampling gridOnly two channelsDouble quincunx sampling = nonseparable samplingLess biased in horizontal and vertical directionsComparable results with separable wavelets

2D Wavelet

Documents

Transcript of 2D Wavelet