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The lifting steps implementation

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Cakewalk construction. x. x. s. s. t. t. dual lifting ... Every finite wvt can be obtained with a cakewalk starting from the Lazy wavelet. Lifting theorem ... – PowerPoint PPT presentation

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Title: The lifting steps implementation


1
The lifting steps implementation
  • Second generation wavelets

2
Introduction
  • The idea of lifting
  • Mathematical context
  • The filter-bank side
  • Polyphase representation
  • Features
  • Conclusions
  • Reference Factoring wavelet transforms into
    lifting steps, I. Daubechies, W. Sweldens, 1996
  • Available on line

Professor, Princeton University
Sr. Vice President of Technology
Commercialization at Lucent Tech. Bell Labs
3
The lifting scheme
  • New philosophy in Biorthogonal wavelet
    construction
  • Sweldens, 95
  • Both linear and non-linear wavelets
  • Integer implementation enabling lossless coding

4
Biorthogonal basiswhy?
  • FIR orthonormal filters no symmetry
  • (except Haar filter)
  • FIR biorthogonal filters symmetry
  • linear phase
  • better boundary conditions

5
Basis oh the Hilbert space
  • Orthonormal basis
  • enn?N family of the Hilbert space
  • lt en, epgt0 ?n?p
  • ?x ? H, ??(n)lt x, engt
  • en21
  • x?n ?(n) en

6
Basis of the Hilbert space
  • Riesz bases
  • enn?N linearly independent
  • ?y ? H, ?Agt0 and Bgt0 y?n ?(n) en
  • y2/B ? ?n ?(n)2 ? y2/A
  • ?(n)lt y, engt
  • enn?N dual family
  • Biorthogonality relationship lt en, epgt?(n-p)
  • y?n lt y, engt en
  • AB1 ? orthogonal basis

7
Biorthogonal filters
h
h
?2
?2
x
x

g
g
?2
?2
8
Rationale
  • Goal Exploit the correlation structure present
    in most real life signals to build a sparse
    approximation
  • The correlation structure is typically local in
    space (time) and frequency
  • Basic idea
  • Split the signal x in its polyphase components
    (even and odd samples)
  • These two are highly correlated. It is thus
    natural to use one of them (e.g. the odds) to
    predict the other (e.g. the even)
  • The operation of computing a prediction and
    recording the detail we call lifting step

predicted
difference or detail
9
Lifting steps
  • To get a good frequency splitting, the evens are
    also updated by replacing them with a smoothed
    version
  • Built-in feature of lifting no matter how P and
    U are chosen, the scheme is always invertible and
    thus leads to critically sampled perfect
    reconstruction filter banks

xo
s

x
split
d

xe
10
Polyphase representation
even coefficients
odd coefficients
Polyphase matrix
PR condition
11
Polyphase representation
  • The problem of finding a FIR wavelet transform
    then amounts to finding a matrix P(z) with
    determinant 1
  • Once the matrix is given, the filters follow
  • One can show that this corresponds to the
    biorthogonality relations

Lazy wavelet
xe
P(z-1)
P(z)
LP
?2
?2
x

HP
?2
?2
z
z-1
xo
12
The lifting scheme
  • Definition 1. A filter pair (h,g) is
    complementary if the corresponding polyphase
    matrix P(z) has determinant 1
  • If (h,g) is complementary, so is
  • Theorem 3 (Lifting). Let (h,g) be complementary.
    Then, any other finite filter gnew(z)
    complementary to h is of the form
  • where s(z) is a Laurent polynomial. Conversely,
    any filter of this form is complementary to h
  • Proof

13
The lifting scheme
  • Correspondingly, at the analysis side

Update
-

14
Towards lossless
LP Xe
-

s(z)
s(z)
HP Xo
do
undo
15
Dual lifting
  • Teorem 4. Let (h,g) be complementary. Then any
    other filter hnew(z) complementary to g is of the
    form
  • where t(z) is a Laurent polynomial. Conversely,
    any filter of this form is complementary to g
  • New polyphase matrix
  • Dual lifting creates a new given by

16
Dual lifting
-

Prediction
  • Prediction steps the HP coefficients are shaped
    (lifted) by filtering the LP ones by the filter
    t(z)
  • Update steps the LP coefficients are shaped by
    filtering the HP ones by s(z)
  • One can start from the lazy wavelet and use
    lifting to gradually build ones way up to a
    multiresolution analysis with particular
    properties

17
Global Lifting
Lifting
Lifting ?
Dual Lifting
Lifting ?
18
Cakewalk construction
lifting (prediction)
?2
?2

-
x
x
s
s
t
t

?2
?2
-

dual lifting (update)
19
Lifting the Lazy wavelet
?2
?2

-
x
x
s
s
t
t

?2
?2
z
z-1
-

Every finite wvt can be obtained with a cakewalk
starting from the Lazy wavelet
20
Lifting theorem
  • Theorem 7. Given a complementary filter pair
    (h,g), then there always exist Laurent
    polynomials si(z) and ti(z) for i1,...,m and a
    non-zero costant K so that
  • The dual polyphase matrix is given by
  • Every finite filter wavelet transform can be
    obtained by starting with the lazy wavelet
    followed by m lifting and dual lifting steps,
    followed by a scaling

21
Implementation
Analysis
Xe
1/K
LP
-
-
X
s1(z)
t1(z)
sm(z)
tm(z)
Xo
K
HP
z
-
-
Synthesis
LP
K


X
s1(z)
t1(z)
sm(z)
tm(z)
HP
1/K
z


22
Lifted basis functions
  • Lifting
  • Dual lifting

does not change ? lifting the wavelet through s(z)
does not change ? lifting the basis function
through t(z)
23
Integer wavelet transform
Analysis
Synthesis
Xe
-

si(z)
si(z)
round
round
X
X
ti(z)
ti(z)
round
round
z-1
z
-

Xo
Lazy wavelet
do
undo
Lossless coding
24
Fully in-place implementation
  • Odd samples are used to predict even samples and
    viceversa
  • The original memory locations can be overwritten

Original
decomposition tree
25
Summary
  • Biorthogonal (FIR) wavelets
  • Faster, fully in-place implementation
  • Reduced computational complexity
  • All operations within one lifting step can be
    done entirely parallel while the only sequential
    part is the order of the lifting operations
  • Allows wavelets mapping integers to integers,
    important for hardware implementation and
    lossless coding
  • Allows for adaptive wavelet transforms (i.e.
    wavelets on the sphere)

26
Application Object-based coding
Object1
Object2
Header
Border dimension
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