2d wavelet
TRANSCRIPT
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Digital Image Processing II
2D Wavelets for Different Sampling
Grids and the Lifting Scheme
Miroslav VrankićUniversity of Zagreb, Croatia
Presented by: Atanas Gotchev
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Digital Image Processing II
Lecture Outline
1D wavelets and FWT2D separable wavelets2D nonseparable wavelets– different sampling grids
Lifting scheme– easy to construct filter banks
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Digital Image Processing II
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:
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Digital Image Processing II
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!
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Digital Image Processing II
FWT: Analysis Filter Bank
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Digital Image Processing II
FWT: Analysis Filter Bank
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Digital Image Processing II
Synthesis Bank
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Digital Image Processing II
Synthesis Bank
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Digital Image Processing II
Complexity of FWT
Number of operations proportional to:N – size of dataL – length of filters in the filterbank (scaling and wavelet vectors)
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Digital Image Processing II
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)
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Digital Image Processing II
2D separable FWT
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Digital Image Processing II
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Digital Image Processing II
Example: Symlets wavelets
See functionssymaux,dbauxin WaveletToolbox
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Digital Image Processing II
Wavelet and the Scaling Function
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Digital Image Processing II
2D wavelets and scaling function
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Digital Image Processing II
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Digital Image Processing II
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...
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Digital Image Processing II
Quincunx Downsampling
n2
n1
Image is split into two phases (cosets)Simplest nonseparable sampling scheme
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Digital Image Processing II
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
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Digital Image Processing II
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.
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
Quincunx Polyphase Decomposition
Phase 2
Phase 1
Counterclockwise rotation
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
Bandlimiting
Properly bandlimited signal for quincunx downsampling
ω1π
πω2
ω2
ω1π 2π
π
2π
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Digital Image Processing II
Quincunx downsampling
Input image has been properly bandlimited
Spectrum support of the downsampled image
ω2
ω1π 2π
π
2π
ω2
ω1π 2π
π
2π
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Digital Image Processing II
Quincunx upsampling
(1,-1)
(1,1)
n2
n1(1,0)
(0,1)
n2
n1
1( ) if ( )( )0 otherwiseU
x LATx−⎧ ∈⎪= ⎨
⎪⎩
D n n Dn
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Digital Image Processing II
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
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Digital Image Processing II
Frequency transformation
ωz je→
ωDDzTj
ddj
ddj
dd
dd
eee
zzzz
=⎥⎦
⎤⎢⎣
⎡→⎥
⎦
⎤⎢⎣
⎡=
+
+
)(
)(
21
21222112
221111
2212
2111
ωω
ωω
)()( ωDω TU XX =Conclusion:
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Digital Image Processing II
Quincunx upsampling
( ) ( )TUX X=ω D ω( )X ω
ω2
ω1π 2π
π
2π
ω2
ω1π 2π
π
2π
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
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
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Digital Image Processing II
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.
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Digital Image Processing II
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
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Digital Image Processing II
Supports of the Prediction Filters
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Digital Image Processing II
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=
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Digital Image Processing II
Transfer Functions for P4 and U2
Synthesis LP
Analysis LP Analysis HP
Synthesis HP
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Digital Image Processing II
Wavelet and Scale for P4 and U2
Analysis wavelet
Synthesis scale
Analysis scale
Synthesis wavelet
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Digital Image Processing II
Wavelet Decomposition Tree
AJ-1
DJ-1
AJ-2
DJ-2
AJ-3
DJ-3
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Digital Image Processing II
Separable Versus Nonseparable
Nonseparable– higher complexity– more freedom in FB design– different directional properties
Separable– widely used– simple realization based on 1D filter banks
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Digital Image Processing II
Quincunx Wavelets
Simplest nonseparable sampling gridOnly two channelsDouble quincunx sampling = nonseparable samplingLess biased in horizontal and vertical directionsComparable results with separable wavelets