liftingScheme

Create lifting scheme for lifting wavelet transform

Since R2021a

Description

Use the liftingScheme object to create a lifting scheme that you can efficiently apply to data.

Creation

Syntax

lscheme = liftingScheme

lscheme = liftingScheme(Name,Value)

Description

lscheme = liftingScheme creates the lifting scheme for the 'lazy' wavelet with normalization set to 1.

lscheme = liftingScheme(Name,Value) creates a lifting scheme with Properties specified by name-value pairs. Enclose the property name in quotes. You can create a lifting scheme using one of the following syntaxes:

• lscheme = liftingScheme('Wavelet',wname)

• lscheme = liftingScheme('CustomLowpassFilter',filter)

• lscheme = liftingScheme('LiftingSteps',liftingSteps,'NormalizationFactors',normFactors)

Properties

Wavelet — Orthogonal or biorthogonal wavelet

'lazy' (default) | 'haar' | 'db1' | 'db2' | ...

Orthogonal or biorthogonal wavelet associated with the lifting scheme, specified as one of these.

Wavelet Family	Wavelet
Daubechies	'lazy', 'haar', 'db1', 'db2', 'db3', 'db4', 'db5', 'db6', 'db7', and 'db8'
Symlet	'sym2', 'sym3', 'sym4', 'sym5', 'sym6', 'sym7', and 'sym8'
Cohen-Daubechies-Feauveau	'cdf1.1', 'cdf1.3', 'cdf1.5', 'cdf2.2', 'cdf2.4', 'cdf2.6', 'cdf3.1', 'cdf3.3', 'cdf3.5', 'cdf4.2', 'cdf4.4', 'cdf4.6', 'cdf5.1', 'cdf5.3', 'cdf5.5', 'cdf6.2', 'cdf6.4', and 'cdf6.6'
Coiflet	'coif1', and 'coif2'
Biorthogonal	'bior1.1', 'bior1.3','bior1.5', 'bior2.2', 'bior2.4', 'bior2.6', 'bior2.8', 'bior3.1', 'bior3.3', 'bior3.5', 'bior3.7', 'bior3.9', 'bior4.4', 'bior5.5', 'bior6.8', 'bs3', and '9.7'
Reverse Biorthogonal	'rbs3', 'r9.7', 'rbio1.1', 'rbio1.3', 'rbio1.5', 'rbio2.2', 'rbio2.4', 'rbio2.6', 'rbio2.8', 'rbio3.1', 'rbio3.3', 'rbio3.5', 'rbio3.7', 'rbio3.9', 'rbio4.4', 'rbio5.5', and 'rbio6.8'

Example: lscheme = liftingScheme('Wavelet','bior3.7') creates the lifting scheme associated with the 'bior3.7' biorthogonal wavelet.

CustomLowpassFilter — Lowpass filters

cell array

Lowpass filters associated with the lifting scheme, specified as a cell array.

• To create a lifting scheme associated with an orthogonal wavelet, set CustomLowpassFilter to {LoD}, where LoD is the lowpass filter associated with wavelet.

• To create a lifting scheme associated with a biorthogonal wavelet, set CustomLowpassFilter to {LoPrimal,LoDual}, where LoPrimal and LoDual are the lowpass filters associated with the biorthogonal wavelet.

When you specify filter coefficients, the Wavelet property is automatically set to 'custom'.

Example: lscheme = liftingScheme('CustomLowpassFilter',{[sqrt(2)/2 sqrt(2)/2]}) creates a lifting scheme associated with the Haar wavelet.

Data Types: single | double

LiftingSteps — Lifting steps

liftingStep structure | array of liftingStep structures

Lifting steps associated with the lifting scheme, specified as an array of structures obtained from liftingStep. To create a lifting scheme using LiftingSteps, you must also set the NormalizationFactors property. When you set these two properties, the Wavelet property is automatically set to 'custom'.

Example: lscheme = liftingScheme('LiftingSteps',ELS,'NormalizationFactors',NF) creates a lifting scheme using the liftingStep structures specified in ELS and factors specified in NF.

NormalizationFactors — Normalization factors

non-zero scalar | vector

Normalization factors associated with the lifting scheme, specified as K or [K 1/K], where K is a non-zero scalar. The factor K specifies the diagonal elements of the 2-by-2 normalization matrix. If specified as a vector, the product of the vector elements must equal 1 to within precision.

To create a lifting scheme using NormalizationFactors, you must also set the LiftingSteps property. When you set these two properties, the Wavelet property is automatically set to 'custom'.

Data Types: double

Object Functions

addlift	Add elementary lifting steps
deletelift	Delete elementary lifting steps
ls2filt	Extract wavelet filters from lifting scheme
disp	Display lifting scheme

Examples

Apply Lifting Scheme to Signal

Create the lifting scheme associated with the Haar wavelet.

lscheme = liftingScheme('Wavelet','haar')

lscheme = 
 	 Wavelet               : 'haar' 
	 LiftingSteps          : [2 × 1] liftingStep 
	 NormalizationFactors  : [1.4142 0.7071] 
	 CustomLowpassFilter   : [  ] 


 Details of LiftingSteps :
            Type: 'predict'
    Coefficients: -1
        MaxOrder: 0

            Type: 'update'
    Coefficients: 0.5000
        MaxOrder: 0

Obtain the level 2 wavelet decomposition of a signal using the lifting scheme. Inspect the approximation and detail coefficients.

sig = 0:7;
[appC,detC]=lwt(sig,'LiftingScheme',lscheme,'Level',2);
appC

appC = 2×1

    3.0000
   11.0000

detC{1}

ans = 4×1

    0.7071
    0.7071
    0.7071
    0.7071

detC{2}

ans = 2×1

    2.0000
    2.0000

Obtain the inverse transform and demonstrate perfect reconstruction.

xrec = ilwt(appC,detC,'LiftingScheme',lscheme);
max(abs(xrec(:)-sig(:)))

ans = 
2.6645e-15

Create Lifting Scheme Using Custom Lowpass Filter

Create a lifting scheme using the lowpass filters associated with the db4 wavelet.

wv = 'db4';
[~,~,LoR,~] = wfilters(wv);
LS = liftingScheme('CustomLowpassFilter',{LoR});

Demonstrate Wavelet Orthogonality

Create the lifting scheme associated with the biorthogonal bior2.2 wavelet.

lscheme = liftingScheme('Wavelet','bior2.2');

A wavelet with $$N$$ vanishing moments is orthogonal to degree $$N-1$$ polynomials. The bior2.2 wavelet has two vanishing moments. Create a signal by sampling a polynomial of degree 1.

sig = 1:16;

Apply the lifting scheme to the signal. Inspect the detail coefficients at the finest scale. The bior2.2 wavelet is orthogonal to the degree 1 polynomial. Confirm that except for the nonzero coefficient at the boundary, the detail coefficients are zero.

[A,D] = lwt(sig,'LiftingScheme',lscheme);
D{1}

ans = 8×1

         0
         0
         0
         0
         0
         0
         0
    5.6569

Now create the lifting scheme associated with the Haar wavelet.

lschemeH = liftingScheme('Wavelet','haar');

Apply the lifting scheme to the signal. Confirm the detail coefficients are all nonzero. Because the Haar wavelet has only one vanishing moment, the wavelet is not orthogonal to the degree 1 polynomial.

[AH,DH] = lwt(sig,'LiftingScheme',lschemeH);
DH{1}

ans = 8×1

    0.7071
    0.7071
    0.7071
    0.7071
    0.7071
    0.7071
    0.7071
    0.7071

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The disp object function is not supported.

Version History

Introduced in R2021a

R2021b: CustomLowpassFilter name-value argument in liftingScheme must be a cell array

Starting this release, to use a lowpass filter to create a lifting scheme associated with an orthogonal wavelet, you must specify CustomLowpassFilter as a cell array. If you specify CustomLowpassFilter as a vector, liftingScheme will generate an error.

To update your code, change instances of 'CustomLowpassFilter',lpass, where lpass is the vector, to 'CustomLowpassFilter',{lpass}.