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imlpt

Inverse multiscale local 1-D polynomial transform

Description

example

y = imlpt(coefs,T,coefsPerLevel,scalingMoments) returns the inverse multiscale local polynomial 1-D transform (MLPT) of coefs. The inputs to imlpt must be the outputs of mlpt.

example

y = imlpt(___,Name,Value) specifies mlpt properties using one or more Name,Value pair arguments and the input arguments from the previous syntax.

Examples

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Create a signal with nonuniform sampling and verify good reconstruction when performing the mlpt and imlpt.

Create and plot a sine wave with non-uniform sampling.

timeVector = 0:0.01:1;
sineWave = sin(2*pi*timeVector)';

samplesToErase = randi(100,100,1);
sineWave(samplesToErase) = [];
timeVector(samplesToErase) = [];

figure(1)
plot(timeVector,sineWave,'o')
hold on

Figure contains an axes object. The axes object contains an object of type line.

Perform the multiscale local 1-D polynomial transform (mlpt) on the signal. Visualize the coefficients.

[coefs,T,coefsPerLevel,scalingMoments] = mlpt(sineWave,timeVector);

figure(2)
stem(coefs)
title('Wavelet Coefficients')

Figure contains an axes object. The axes object with title Wavelet Coefficients contains an object of type stem.

Perform the inverse multiscale local 1-D polynomial transform (imlpt) on the coefficients. Visualize the reconstructed signal.

y = imlpt(coefs,T,coefsPerLevel,scalingMoments);

figure(1)
plot(T,y,'*')
legend('Original Signal','Reconstructed Signal')
hold off

Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Original Signal, Reconstructed Signal.

Look at the total error to verify good reconstruction.

reconstructionError = sum(abs(y-sineWave))
reconstructionError = 2.8383e-15

Specify nondefault dual moments by using the mlpt function. Compare the results of analysis and synthesis using the default and nondefault dual moments.

Create an input signal and visualize it.

T = (1:16)';
x = T.^2;
plot(x)
hold on

Figure contains an axes object. The axes object contains an object of type line.

Perform the forward and inverse transform for the input signal using the default and nondefault dual moments.

[w2,t2,nj2,scalingmoments2] = mlpt(x,T);
y2 = imlpt(w2,t2,nj2,scalingmoments2);

[w3,t3,nj3,scalingmoments3] = mlpt(x,T,'dualmoments',3);
y3 = imlpt(w3,t3,nj3,scalingmoments3,'dualmoments',3);

Plot the reconstructed signal and verify perfect reconstruction using both the default and nondefault dual moments.

plot(y2,'o')
plot(y3,'*')
legend('Original Signal', ...
       'DualMoments = 3', ...
       'DualMoments = 2 (Default)');

fprintf('\nMean Reconstruction Error:\n');
Mean Reconstruction Error:
fprintf('  - Nondefault dual moments: %0.2f\n',mean(abs(y3-x)));
  - Nondefault dual moments: 0.00
fprintf('  - Default dual moments: %0.2f\n\n',mean(abs(y2-x)));
  - Default dual moments: 0.00
hold off

Figure contains an axes object. The axes object contains 3 objects of type line. These objects represent Original Signal, DualMoments = 3, DualMoments = 2 (Default).

Input Arguments

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MLPT coefficients, specified as a vector or matrix of MLPT coefficients returned by the mlpt function.

Data Types: double

Sampling instants corresponding to y, specified as a vector or duration array of increasing values returned by the mlpt function.

Data Types: double | duration

Coefficients per resolution level, specified as a vector containing the number of coefficients at each resolution level in coefs. coefsPerLevel is an output argument of the mlpt function.

The elements of coefsPerLevel are organized as follows:

  • coefsPerLevel(1) — Number of approximation coefficients at the coarsest resolution level.

  • coefsPerLevel(i) — Number of detail coefficients at resolution level i, where i = numLevel – i + 2 for i = 2, ..., numLevel + 1. numLevel is the number of resolution levels used to calculate the MLPT. numLevel is inferred from coefsPerLevel: numLevel = length(coefsPerLevel-1).

The smaller the index i, the lower the resolution. The MLPT is two times redundant in the number of detail coefficients, but not in the number of approximation coefficients.

Data Types: double

Scaling function moments, specified as a length(coefs)-by-P matrix, where P is the number of primal moments specified by the MLPT.

Data Types: double

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'DualMoments',3 computes a transform using three dual vanishing moments.

Number of dual vanishing moments in the lifting scheme, specified as the comma-separated pair consisting of 'DualMoments' and 2, 3 or 4. The number of dual moments must match the number used by mlpt.

Data Types: double

Output Arguments

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Reconstructed signal, returned as a vector or matrix, depending on the inputs to the mlpt function.

Data Types: double

Algorithms

Maarten Jansen developed the theoretical foundation of the multiscale local polynomial transform (MLPT) and algorithms for its efficient computation [1][2][3]. The MLPT uses a lifting scheme, wherein a kernel function smooths fine-scale coefficients with a given bandwidth to obtain the coarser resolution coefficients. The mlpt function uses only local polynomial interpolation, but the technique developed by Jansen is more general and admits many other kernel types with adjustable bandwidths [2].

References

[1] Jansen, Maarten. “Multiscale Local Polynomial Smoothing in a Lifted Pyramid for Non-Equispaced Data.” IEEE Transactions on Signal Processing 61, no. 3 (February 2013): 545–55. https://doi.org/10.1109/TSP.2012.2225059.

[2] Jansen, Maarten, and Mohamed Amghar. “Multiscale Local Polynomial Decompositions Using Bandwidths as Scales.” Statistics and Computing 27, no. 5 (September 2017): 1383–99. https://doi.org/10.1007/s11222-016-9692-8.

[3] Jansen, Maarten, and Patrick Oonincx. Second Generation Wavelets and Applications. London ; New York: Springer, 2005.

Introduced in R2017a