The so-called *first generation* wavelets
and scaling functions are dyadic dilations and translates of a single
function. Fourier methods play a key role in the design of these wavelets.
However, the requirement that the wavelet basis consist of translates
and dilates of a single function imposes some constraints that limit
the utility of the multiresolution idea at the core of wavelet analysis.

The utility of wavelet methods is extended by the design of *second
generation* wavelets via *lifting*.

Typical settings where translation and dilation of a single function cannot be used include:

*Designing wavelets on bounded domains*— This includes the construction of wavelets on an interval, or bounded domain in a higher-dimensional Euclidean space.*Weighted wavelets*— In certain applications, such as the solution of partial differential equations, wavelets biorthogonal with respect to a weighted inner product are needed.*Irregularly-spaced data*— In many real-world applications, the sampling interval between data samples is not equal.

Designing new *first generation* wavelets
requires expertise in Fourier analysis. The lifting method proposed
by Sweldens (see [Swe98] in References) removes the necessity of expertise in
Fourier analysis and allows you to generate an infinite number of
discrete biorthogonal wavelets starting from an initial one. In addition
to generation of *first generation* wavelets with
lifting, the lifting method also enables you to design *second
generation* wavelets, which cannot be designed using Fourier-based
methods. With lifting, you can design wavelets that address the shortcomings
of the *first generation* wavelets.

The following section introduces the theory behind lifting, presents the lifting functions of Wavelet Toolbox™ software and gives two short examples:

For more information on lifting, see [Swe98], [Mal98], [StrN96], and [MisMOP03] in References.

The DWT implemented by a filter bank is defined by four filters as described in Fast Wavelet Transform (FWT) Algorithm. Two main properties of interest are

The perfect reconstruction property

The link with “true” wavelets (how to generate, starting from the filters, orthogonal or biorthogonal bases of the space of the functions of finite energy)

To illustrate the perfect reconstruction property, the following filter bank contains two decomposition filters and two synthesis filters. The decomposition and synthesis filters may constitute a pair of biorthogonal bases or an orthogonal basis. The capital letters denote the Z-transforms of the filters..

This leads to the following two conditions for a perfect reconstruction (PR) filter bank:

$$\stackrel{~}{H}(z)H(z)+\stackrel{~}{G}(z)G(z)=2{z}^{-L+1}$$

and

$$\stackrel{~}{H}(-z)H(z)+\stackrel{~}{G}(-z)G(z)=0$$

The first condition is usually (incorrectly) called the perfect reconstruction condition and the second is the anti-aliasing condition.

The *z ^{–L+1}* term
implies that perfect reconstruction is achieved up to a delay of one
sample less than the filter length,

*Lifting* designs perfect reconstruction
filter banks by beginning from the basic nature of the wavelet transform.
Wavelet transforms build sparse representations by exploiting the
correlation inherent in most real world data. For example, plot the
example of electricity consumption over a 3-day period.

load leleccum; plot(leleccum) grid on; axis tight;

The
data do not exhibit arbitrary changes from sample to sample. Neighboring
samples exhibit correlation. A relatively low (high) value at index
(sample) *n* is associated with a relatively low
(high) value at index *n-1* and *n+1*.
This implies that if you have only the odd or even samples from the
data, you can predict the even or odd samples. How accurate your prediction
is obviously depends on the nature of the correlation between adjacent
samples and how closely your predictor approximates that correlation.

The *polyphase* representation of a signal
is an important concept in lifting. You can view each signal as consisting
of *phases*, which consist of taking every *N*-th
sample beginning with some index. For example, if you index a time
series from *n*=0 and
take every other sample starting at *n*=0,
you extract the even samples. If you take every other sample starting
from *n*=1,
you extract the odd samples. These are the even and odd polyphase
components of the data. Because your increment between samples is
2, there are only two phases. If you increased your increment to 4,
you can extract 4 phases. For lifting, it is sufficient to concentrate
on the even and odd polyphase components. The following diagram illustrates
this operation for an input signal.

where *Z* denotes
the unit advance operator and the downward arrow with the number 2
represents downsampling by two. In the language of lifting, the operation
of separating an input signal into even and odd components is known
as the *split* operation, or the *lazy
wavelet*.

To understand lifting mathematically, it is necessary to understand the z-domain representation of the even and odd polyphase components.

The z-transform of the even polyphase component is

$${X}_{0}(z)={\displaystyle \sum _{n}x}(2n){z}^{-n}$$

The z-transform of the odd polyphase component is

$${X}_{1}(z)={\displaystyle \sum _{n}x}(2n+1){z}^{-n}$$

You can write the z-transform of the input signal as the sum of dilated versions of the z-transforms of the polyphase components.

$$X(z)={\displaystyle \sum _{n}x}(2n){z}^{-2n}+{\displaystyle \sum _{n}x}(2n+1){z}^{-2n+1}={X}_{0}({z}^{2})+{z}^{-1}{X}_{1}({z}^{2})$$

A single lifting step can be described by the following three basic operations:

*Split*— the signal into disjoint components. A common way to do this is to extract the even and odd polyphase components explained in Polyphase Representation. This is also known as the*lazy wavelet*.*Predict*— the odd polyphase component based on a linear combination of samples of the even polyphase component. The samples of the odd polyphase component are replaced by the difference between the odd polyphase component and the predicted value. The predict operation is also referred to as the*dual lifting step*.*Update*— the even polyphase component based on a linear combination of difference samples obtained from the predict step. The update step is also referred to as the*primal lifting step*.

In practice, a normalization is incorporated for both the primal and dual liftings.

The following diagram illustrates one lifting step.

Using the operations defined in Split, Predict, and Update, you can implement the Haar wavelet via lifting.

*Split*— Divide the signal into even and odd polyphase components.*Predict*— Replace*x(2n+1)*with*d(n)*=*x(2n+1)*–*x(2n)*. The predict operator is simply*x(2n)*.*Update*— Replace*x(2n)*with*x(2n)*+*d(n)/2*. This is equal to (*x(2n)*+*x(2n+1)*)/2.

The dual lifting in the z domain can be written in the following matrix form

$$\left(\begin{array}{cc}1& 0\\ -P(z)& 1\end{array}\right)\left(\begin{array}{c}{X}_{0}(z)\\ {X}_{1}(z)\end{array}\right)$$

with *P(z)*=1.

The primal lifting can be written in the z domain in the following matrix form

$$\left(\begin{array}{cc}1& S(z)\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ -P(z)& 1\end{array}\right)\left(\begin{array}{c}{X}_{0}(z)\\ {X}_{1}(z)\end{array}\right)$$

with *S(z)*=1/2.

Finally, the primal and dual normalization can be incorporated as follows.

$$\left(\begin{array}{cc}\sqrt{2}& 0\\ 0& \frac{1}{\sqrt{2}}\end{array}\right)\left(\begin{array}{cc}1& S(z)\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ -P(z)& 1\end{array}\right)\left(\begin{array}{c}{X}_{0}(z)\\ {X}_{1}(z)\end{array}\right)$$

To construct this lifting step in MATLAB^{®}, enter:

```
LiftHaar = liftwave('haar');
displs(LiftHaar)
```

The following is displayed in the MATLAB command window.

LiftHaar = {... 'd' [ -1.00000000] [0] 'p' [ 0.50000000] [0] [ 1.41421356] [ 0.70710678] [] };

`'d'`

denotes the dual lifting. Note that for
convenience, the negative sign is incorporated into the dual lifting
step in the Wavelet
Toolbox software. `'p'`

denotes
the primal lifting and `[ 1.41421356] [ 0.70710678]`

are
the primal and dual normalization constants. `LiftHaar{1,3}`

and `LiftHaar{2,3}`

give
the highest degree of the Laurent polynomials, which describe the
dual and primal liftings. In this case, both are zero because the
dual and primal liftings are both described by scalars.

This example presents the lifting scheme for the `bior2.2`

biorthogonal
scaling and wavelet filters.

In the Haar lifting scheme, the dual lifting (predict operator) differenced the odd and even samples. In this example, define a new predict operator that computes the average of the two neighboring even samples. Subtract the average from the intervening odd sample.

$$d(n)=x(2n+1)-\frac{1}{2}[x(2n)+x(2n+2)]$$

In the z-domain you can write the dual lifting step as

$$\left(\begin{array}{cc}1& 0\\ -\frac{1}{2}(1+z)& 1\end{array}\right)\left(\begin{array}{c}{X}_{0}(z)\\ {X}_{1}(z)\end{array}\right)$$

To obtain the primal lifting, or update, examine the primal lifting in Haar Wavelet Via Lifting. The update is defined in such a way that the sum of the approximation coefficients is proportional to the mean of the input data vector.

$$\sum _{n}x}(n)=\frac{1}{2}{\displaystyle \sum _{n}a}(n)$$

To obtain the same result in this lifting step, define the update as

$$\left(\begin{array}{cc}1& \frac{1}{4}({z}^{-1}+1)\\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& 0\\ -\frac{1}{2}(1+z)& 1\end{array}\right)\left(\begin{array}{c}{X}_{0}(z)\\ {X}_{1}(z)\end{array}\right)$$

To obtain this lifting scheme at the command line, enter:

`liftwave('bior2.2')`

The lifting functions of the toolbox are organized into five groups:

These functions connect lifting schemes to biorthogonal quadruplets of filters and associated scaling and wavelet function pairs.

Function Name | Description |
---|---|

Apply elementary lifting steps on quadruplet of filters | |

Transform a quadruplet of filters to a lifting scheme | |

Transform a lifting scheme to a quadruplet of filters | |

Compute and plot biorthogonal “scaling and wavelet” functions |

These functions provide some basic lifting schemes associated with some usual orthogonal or biorthogonal (“true”) wavelets and the “lazy” one. These schemes can be used to initialize a lifting procedure.

These functions contain the direct and inverse lifting wavelet transform (LWT) files for both 1-D and 2-D signals. LWT reduces to the polyphase version of the DWT algorithm with zero-padding extension mode and without extra-coefficients.

These functions permit an entry to representation and calculus of Laurent polynomials and matrices.

Function Name | Description |
---|---|

Constructor for the class of Laurent polynomials | |

Constructor for the class of Laurent matrices |

The lifting folder and the two object folders `@laurpoly`

and `@laurmat`

contain
many other files.

These two simple examples illustrate the basic lifting capabilities of Wavelet Toolbox software.

A primal lifting starting from Haar wavelet.

Start from the Haar wavelet and get the corresponding lifting scheme.

```
lshaar = liftwave('haar');
displs(lshaar);
```

Add a primal ELS to the lifting scheme.

```
els = {'p',[-0.125 0.125],0};
lsnew = addlift(lshaar,els);
displs(lsnew);
```

Transform the lifting scheme to biorthogonal filters quadruplet and plot the resulting scaling function and wavelet.

```
[LoD,HiD,LoR,HiR] = ls2filt(lsnew);
bswfun(LoD,HiD,LoR,HiR,'plot');
```

In several applications it is desirable to have a wavelet transform that maps integer inputs to integer scaling and wavelet coefficients. You can accomplish easily using lifting.

Start with the Haar transform for an integer to integer wavelet transform and apply a primal lifting step.

lshaar = liftwave('haar','int2int'); els = {'p',[-0.125 0.125],0}; lsnewint = addlift(lshaar,els);

Obtain the integer-to-integer wavelet transform of a 1-D signal and invert the transform to demonstrate perfect reconstruction.

x = 1:8; [cA,cD] = lwt(x,lsnewint); xnew = ilwt(cA,cD,lsnewint)