compute the inverse discrete wavelet transform of data - Maple Help

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DiscreteTransforms[InverseDiscreteWaveletTransform] - compute the inverse discrete wavelet transform of data

Calling Sequence

InverseDiscreteWaveletTransform(A1, A2, w1, w2)

InverseDiscreteWaveletTransform(A1, A2, w1, w2, options)

InverseDiscreteWaveletTransform(A1, A2, Wavelet, WaveletParams, options)

InverseDiscreteWaveletTransform(n, A1, A2, dim, w1, w1, A, options)

InverseDiscreteWaveletTransform(n, m, A1, A2, dim, w1, w1, A, options)

InverseDiscreteWaveletTransform(n, A1, A2, dim, Wavelet, WaveletParams, A, options)

InverseDiscreteWaveletTransform(n, m, A1, A2, dim, Wavelet, WaveletParams, A, options)

Parameters

A1, A2

-

Vector, Matrix, or Array of datatype=float[8]; the data to be inverse transformed

w1, w2

-

Vectors of datatype=float[8] of the same length; the high and low pass synthesis filters

Wavelet

-

string or name; the name of a supported wavelet family

WaveletParams

-

positive integer or list of two positive integers; the parameters to the family Wavelet

n

-

(optional) positive integer; the number of data points transformed in the original data

dim

-

(optional) integer; the dimension to be inverse transformed in A, if A has more than one dimension

A

-

(optional) Vector, Matrix, or Array of datatype=float[8]; an rtable in which to store the inverse transformed data

m

-

(optional) positive integer; the number of inverse transforms to be done

options

-

(optional) equation(s) of the form keyword = value, where keyword is endconditions or storagetype

Description

• 

The InverseDiscreteWaveletTransform command inverts DiscreteWaveletTransform on data A1 and A2. If A is provided, the inverse transformed data is stored in it and it is returned. Otherwise, A is created before it is returned.

• 

If A is created, then it is created to be of the same type, and have the same options, as A1.

• 

If provided, Wavelet and WaveletParams must be supported by WaveletCoefficients.

• 

If Wavelet is an orthogonal family, then

  

InverseDiscreteWaveletTransform(A, Wavelet, WaveletParams)

  

is equivalent to

  

InverseDiscreteWaveletTransform(A, WaveletCoefficients(Wavelet,WaveletParams))

• 

If Wavelet is a biorthogonal family, then

  

InverseDiscreteWaveletTransform(A, Wavelet, WaveletParams)

  

is equivalent to

  

InverseDiscreteWaveletTransform(A, op(1..2,[WaveletCoefficients(Wavelet,op(WaveletParams))]))

• 

If A has more than one dimension, dim must be provided.

• 

Note that n is the size of the original data, not the transformed data. Specifying n indicates that n points were transformed, so if storagetype=standard, then n/2 points in each of A1 and A2 are considered in the inverse.

• 

If A1 and A2 have more than one dimension, specifying m potentially stops InverseDiscreteWaveletTransform from transforming all the data in A1 and A2 along dimension dim. Instead, only m inverse transforms are performed. So, for example, if A1 and A2 are Matrices and m=1, then only the first row or column of A1 and A2 is inverse transformed.  

• 

m can only be specified if n is also provided.

• 

Note: Currently, only Arrays of 1 or 2 dimensions can be transformed.  

Examples

withDiscreteTransforms

DiscreteWaveletTransform,FourierTransform,InverseDiscreteWaveletTransform,InverseFourierTransform,WaveletCoefficients,WaveletPlot

(1)

V:=Vector10,i→i,datatype=float8:

w1,w2:=WaveletCoefficientsDaubechies,4:

V1,V2:=DiscreteWaveletTransformV,w1,w2:

InverseDiscreteWaveletTransformV1,V2,w1,w2

1.2.000000000000003.000000000000004.000000000000005.000000000000006.000000000000007.000000000000008.9.10.0000000000000

(2)

V1:=DiscreteWaveletTransformV,w1,w2,storagetype=singlearray:

InverseDiscreteWaveletTransformV1,w1,w2,storagetype=singlearray

1.2.000000000000003.000000000000004.000000000000005.000000000000006.000000000000007.000000000000008.9.10.0000000000000

(3)

Here is an example of how the columns and then the rows of a Matrix can be transformed and then inverted, using a biorthogonal wavelet. To illustrate the use of the n and m parameters, transform a 6 by 6 block of a 7 by 7 Matrix.

M:=Matrix7,7,datatype=float8,i,j→i+10j:

temp:=Matrix7,7,datatype=float8:

T:=Matrix7,7,datatype=float8:

w1a,w2a,w1s,w2s:=WaveletCoefficientsBiorthogonalSpline,2,2:

DiscreteWaveletTransform6,6,M,1,w1a,w2a,temp,storagetype=singlearray:

DiscreteWaveletTransform6,6,temp,2,w1a,w2a,T,storagetype=singlearray:

InverseDiscreteWaveletTransform6,6,T,2,w1s,w2s,temp,storagetype=singlearray:

InverseDiscreteWaveletTransform6,6,temp,1,w1s,w2s,T,storagetype=singlearray:

MatrixT

11.21.000000000000031.000000000000041.51.61.0.12.000000000000022.000000000000032.000000000000042.52.62.0.13.000000000000023.000000000000033.000000000000043.53.000000000000063.0.14.000000000000024.000000000000034.44.54.64.0.15.25.35.45.000000000000055.65.0.16.26.36.000000000000046.000000000000056.66.0.0.0.0.0.0.0.0.

(4)

See Also

DiscreteWaveletTransform, SignalProcessing[InverseDWT], Wavelet Examples and Applications, WaveletCoefficients, WaveletPlot, Wavelets


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