cwtLayer

Continuous wavelet transform (CWT) layer

Since R2022b

Description

A CWT layer computes the CWT of the input. Use of this layer requires Deep Learning Toolbox™.

Creation

Syntax

layer = cwtLayer

layer = cwtLayer(Name=Value)

Description

layer = cwtLayer creates a CWT layer for a signal of length 1024 samples. The layer uses 10 wavelet filters per octave and periodic boundary conditions. By default, the layer uses the analytic Morse (3,60) wavelet.

The input to cwtLayer must be a real-valued dlarray (Deep Learning Toolbox) object in "CBT" format. The size along the time dimension of the tensor input is padded to equal the value of SignalLength. By default, cwtLayer formats the output as "SCBT". For more information, see Layer Output Format.

Note

cwtLayer initializes the weights internally to be the wavelet filters used in the CWT. Initializing the weights directly is not recommended.

layer = cwtLayer(Name=Value) creates a CWT layer with properties specified by one or more name-value arguments. For example, layer = cwtLayer(Wavelet="amor") creates a layer that uses the analytic Morlet wavelet in the CWT. You can specify multiple name-value arguments.

Properties

CWT

SignalLength — Signal length in samples

1024 (default) | positive integer

This property is read-only.

Signal length in samples, specified as a positive integer greater than or equal to 4. All sequence inputs to cwtLayer are padded to have size SignalLength along the time dimension.

Data Types: single | double

Wavelet — Analysis wavelet

"Morse" (default) | "amor" | "bump"

This property is read-only.

Analysis wavelet used in the CWT, specified as "Morse", "amor", or "bump", representing the analytic Morse, Morlet (Gabor), and bump wavelet, respectively. The default wavelet is the analytic Morse (3,60) wavelet.

Threshold — Weight threshold

1e–8 (default) | positive real scalar

This property is read-only.

Weight threshold, specified as a positive real scalar. cwtLayer uses the threshold value to determine the significant values for each of the CWT filters in the wavelet filter bank prior to any weight modification through learning. It sets values below the specified threshold to zero and excludes them from learning. The CWT filters are normalized so that the peak value is 2 for each filter.

• Smaller values of Threshold result in more values being retained from the CWT filters and therefore less weight reduction.

• Larger values of Threshold result in more weight reduction and more divergence between the deep learning CWT and transforms computed with the full filter bank.

• A threshold less than realmin, the smallest positive normalized floating-point number in double precision, is clipped to realmin for computing significant filter values.

Setting Threshold to a value which results in no values being retained for any individual filter results in an error.

Data Types: single | double

IncludeLowpass — Include lowpass filter

false or 0 (default) | true or 1

This property is read-only.

Include lowpass filter, specified as a numeric or logical 1 (true) or 0 (false). Specify true to include the lowpass (scaling) filter in the CWT.

Data Types: logical

VoicesPerOctave — Number of voices per octave

10 (default) | integer between 1 and 48

This property is read-only.

Number of voices per octave in the CWT, specified as an integer between 1 and 48. The CWT scales are discretized using the specified number of voices per octave. The energy spread of the wavelet in frequency and time automatically determines the minimum and maximum scales.

You can use cwtfreqbounds to determine the frequency limits of the wavelet filter bank. The frequency limits depend on parameters such as the energy spread of the wavelet, number of voices per octave, and signal length.

Data Types: single | double

FrequencyLimits — Frequency limits

two-element scalar vector

This property is read-only.

Frequency limits for the CWT, specified as a two-element vector with positive strictly increasing entries. The frequency limits are interpreted as normalized frequencies, cycles/sample.

• The first element specifies the lowest peak passband frequency and must be greater than or equal to the product of the wavelet peak frequency in normalized frequency and two time standard deviations divided by SignalLength.

• The second element specifies the highest peak passband frequency and must be less than or equal to the Nyquist frequency.

The base-2 logarithm of the ratio of the upper frequency limit freqMax to the lower frequency limit freqMin must be greater than or equal to 1/VoicesPerOctave:

log2(freqMax/freqMin) ≥ 1/VoicesPerOctave.

For more information, see CWT Frequency Limits.

To obtain normalized frequencies, divide your desired frequency limits in hertz by the sample rate in hertz. For example, if the sample rate is 1000 Hz and your desired frequency limits are [100,400] Hz, divide each element by 1000 to obtain the normalized frequencies: [100/1000,400/1000].

Frequency limits apply only to wavelet filters. If you additionally specify IncludeLowpass as true, cwtLayer also includes the lowpass (scaling) filter.

Note

If you use Deep Network Designer (Deep Learning Toolbox) to create or edit a deep learning network, and change the Wavelet property of a cwtLayer, the app does not change the frequency limits of the layer. If you want the app to provide the default frequency limits appropriate for the new wavelet, you must take additional steps. For more information, see Reset Frequency Limits to Default Values in Deep Network Designer.

Data Types: single | double

TimeBandwidth — Time-bandwidth product for Morse wavelet

60 (default) | positive scalar

This property is read-only.

Time-bandwidth product for the Morse wavelet, specified as a positive scalar greater than or equal to 3 and less than or equal to 120. The symmetry (gamma) of the Morse wavelet is fixed at 3. This property is only valid when Wavelet is "Morse". The time-bandwidth product is ignored for the "amor" and "bump" wavelets. For Morse wavelets, the larger the time-bandwidth product, the more spread out the wavelet is in time and narrower the wavelet is in frequency.

In the notation of Morse Wavelets, TimeBandwidth is P².

Data Types: single | double

TransformMode — Layer transform mode

"mag" (default) | "squaremag" | "realimag"

Layer transform mode, specified as one of these:

• "mag" — CWT magnitude

• "squaremag" — CWT squared magnitude

• "realimag" — CWT real and imaginary parts concatenated along the channel dimension

Layer

WeightLearnRateFactor — Multiplier for weight learning rate

0 (default) | nonnegative scalar

Multiplier for weight learning rate, specified as a nonnegative scalar. The weights are the reduced CWT filter values represented as a 1-by-1-by-Nr tensor. See cwtfilters2array for details. By default, the weights do not update with training.

Data Types: single | double

Name — Layer name

"" (default) | character vector | string scalar

Layer name, specified as a character vector or a string scalar. For Layer array input, the trainnet (Deep Learning Toolbox) and dlnetwork (Deep Learning Toolbox) functions automatically assign names to layers with the name "".

The cwtLayer object stores this property as a character vector.

Data Types: char | string

NumInputs — Number of inputs

1 (default)

This property is read-only.

Number of inputs to the layer, returned as 1. This layer accepts a single input only.

Data Types: double

InputNames — Input names

{'in'} (default)

This property is read-only.

Input names, returned as {'in'}. This layer accepts a single input only.

Data Types: cell

NumOutputs — Number of outputs

1 (default)

This property is read-only.

Number of outputs from the layer, returned as 1. This layer has a single output only.

Data Types: double

OutputNames — Output names

{'out'} (default)

This property is read-only.

Output names, returned as {'out'}. This layer has a single output only.

Data Types: cell

Object Functions

filterbank

Full-weight CWT filter bank for deep learning

Examples

Use cwtLayer in Deep Learning Network

Create a CWT layer for a signal of length 2000 samples. Set the learning rate factor to 1.

cLayer = cwtLayer(SignalLength=2000,WeightLearnRateFactor=1);

Create a three-layer dlnetwork containing a sequence input layer, the CWT layer you just made, and a 2-D max pooling layer.

sqLayer = sequenceInputLayer(1,Name="input",MinLength=2000);
mpLayer = maxPooling2dLayer([2 25],Stride=[2 12]);
layers = [sqLayer
    cLayer
    mpLayer];
dlnet = dlnetwork(layers);

Run a batch of 10 random single-channel signals through the dlnetwork.

dataout = forward(dlnet, ...
    dlarray(randn(1,10,2000,"single"),"CBT"));
size(dataout)

ans = 1×4

    40     1    10   165

dims(dataout)

ans = 
'SCBT'

Compare CWT With cwtLayer and Filter Bank

Load the Espiga3 EEG dataset. The data consists of 23 channels of EEG sampled at 200 Hz. There are 995 samples in each channel. Save the multisignal as a dlarray, specifying the dimensions in order. dlarray permutes the array dimensions to the "CBT" shape expected by a deep learning network.

load Espiga3
[N,nch] = size(Espiga3);
x = dlarray(Espiga3,"TCB");
whos Espiga3 x

  Name           Size                Bytes  Class      Attributes

  Espiga3      995x23               183080  double               
  x             23x1x995            183110  dlarray              

Create a CWT filter bank that is appropriate for the channels in the dataset. Use the default analytic Morse (3,60) wavelet. Specify periodic boundary conditions. Use the filter bank to obtain the CWT of one of the channels. The CWT is a 2-D matrix. The row dimension corresponds to scale, or frequency, and the column dimension corresponds to time.

fb = cwtfilterbank(SignalLength=N,Boundary="periodic");
colInd = 11;
cfs = wt(fb,Espiga3(:,colInd));

Layer Transform Mode — "mag"

Create a CWT layer that can be used with the EEG data. By default, the layer outputs the absolute value of the CWT, or scalogram, of each channel.

clayer = cwtLayer(SignalLength=N);

Create a two-layer dlnetwork object containing a sequence input layer and the CWT layer you just created. Treat each channel as a feature. Specify the signal length as the minimum sequence length for the input layer.

slayer = sequenceInputLayer(nch,MinLength=N);
layers = [slayer
    clayer];
dlnet = dlnetwork(layers);

Run the EEG data through the forward method of the network.

dataout = forward(dlnet,x);

By default, the output of cwtLayer is a dlarray object in "SCBT" format. The spatial dimension corresponds to frequency. Convert the network output to a numeric array. Permute the dimensions of the network output to correspond with "STCB" format. The result is a 3-D numeric array because there is only one batch.

q = extractdata(dataout);
q = permute(q,[1 4 2 3]);
whos q cfs

  Name       Size                  Bytes  Class     Attributes

  cfs       71x995               1130320  double    complex   
  q         71x995x23            6499340  single              

Extract from the output the result that corresponds to the channel you chose. Compare with the absolute value of the CWT you obtained previously.

r = q(:,:,colInd);
d = max(abs(r(:)-abs(cfs(:))));
str = sprintf("Difference: %g",d);
fprintf("%s\n",str)

Difference: 8.86966e-06

subplot(2,1,1)
imagesc(r)
title("Layer Output")
subplot(2,1,2)
imagesc(abs(cfs))
title("Filter Bank Output")

Layer Transform Mode — "realimag"

Create a CWT layer that can be used with the data. Specify the layer outputs to be the real and imaginary parts of the CWT. Create a two-layer dlnetwork object containing a sequence input layer and the CWT layer. Run the EEG data through the forward method of the network.

clayer2 = cwtLayer(SignalLength=N,TransformMode="realimag");
layers2 = [slayer
    clayer2];
dlnet2 = dlnetwork(layers2);
dataout2 = forward(dlnet2,x);

Convert the network output to a numeric array. Permute the dimensions of the network output to correspond with "STCB" format. Because the output are the real and imaginary parts of the CWT, the size of the channel dimension is twice the number of input channels.

q2 = extractdata(dataout2);
q2 = permute(q2,[1 4 2 3]);
whos q2

  Name       Size                   Bytes  Class     Attributes

  q2        71x995x46            12998680  single              

Choose a channel. Use the CWT filter bank to obtain the CWT of that channel. Compare the real and imaginary parts of the CWT with the corresponding results from the network output.

colInd = 23;
cfs = wt(fb,Espiga3(:,colInd));
r = q2(:,:,[colInd nch+colInd]);

a1 = real(cfs);
a2 = r(:,:,1);
str1 = sprintf("Difference (real part): %g",max(abs(a1(:)-a2(:))));

a1 = imag(cfs);
a2 = r(:,:,2);
str2 = sprintf("Difference (imag part): %g",max(abs(a1(:)-a2(:))));

fprintf("%s\n%s\n",str1,str2)

Difference (real part): 7.102e-05
Difference (imag part): 6.94453e-05

figure
subplot(2,2,1)
imagesc(real(cfs))
title("Filter Bank — Real")
subplot(2,2,2)
imagesc(r(:,:,1))
title("Layer — Real")
subplot(2,2,3)
imagesc(imag(cfs))
title("Filter Bank — Imag")
subplot(2,2,4)
imagesc(r(:,:,2))
title("Layer — Imag")

CWT Frequency Limits

Create a CWT layer using default values. By default, the CWT layer is for a signal length of 1024 samples. The layer uses the analytic Morse (3,60) wavelet in the CWT with 10 voices per octave and periodic boundary conditions. Inspect the default frequency limits of the layer. The frequency limits, which are in units of cycles/sample, are based on the energy spread of the wavelet, the signal length, and the voices per octave.

clayer = cwtLayer;
clayer.FrequencyLimits

ans = 1×2

    0.0032    0.4341

Use the filterbank method of the layer to obtain the full-weight CWT filter bank for the layer. Each row contains the values of a filter. The filters are ordered from high center frequency to low.

psif = filterbank(clayer);
whos psif

  Name       Size               Bytes  Class     Attributes

  psif      71x1024            290816  single              

You can determine the center frequencies by creating a cwtfilterbank object, specifying the same wavelet parameters as used in the CWT layer, and then use the centerFrequencies object function.

Create a cwtfilterbank. The default wavelet parameters are identical to those in the CWT layer. Obtain the center frequencies. The frequencies are ordered from high to low. Plot the center frequencies.

fb = cwtfilterbank;
cf = centerFrequencies(fb);
subplot(2,1,1)
plot(cf)
ylabel("Cycles/Sample")
title("Wavelet Center Frequencies")
grid on
subplot(2,1,2)
semilogy(cf)
grid on
ylabel("Cycles/Sample")
title("Semi-Log Scale")

The plots show that the wavelet center frequencies are not linearly spaced as is commonly the case with other filter banks. Specifically, the center frequencies are exponentially decreasing. In continuous wavelet analysis, the center frequencies of the wavelet filters are logarithmically spaced. The most common spacing is the base 2^(1/NV), where NV is the number of voices per octave, raised to integer powers. In other words, in a CWT filter bank, it is not possible for consecutive center frequencies $$f_1$$ and $$f_2$$, where $$f_1<f_2$$, to satisfy $${\log }_2(f_2/f_1)<1/NV$$. Frequency limits you specify in the CWT layer must satisfy $${\log }_2(f_2/f_1)\ge 1/NV$$.

The CWT layer uses 10 voices per octave. Compute the base-2 logarithm of the ratios of consecutive pairs of center frequencies. Confirm the minimum and maximum values of the result are both equal to 1/10.

cfRatio = log2(cf(1:end-1)./cf(2:end));
[min(cfRatio) max(cfRatio)]

ans = 1×2

    0.1000    0.1000

Plot the full-weight CWT filter bank and the center frequencies. The center frequencies correspond to the peaks of the frequency response of each wavelet in the filter bank.

slen = clayer.SignalLength;
f = 0:1/slen:1-1/slen;
figure
plot(f,psif')
xlim([0 1/2])
xlabel("Cycles/Sample")
ylabel("Magnitude")
title(["Full-Weight Filter Bank", "With Center Frequencies"])
hold on
plot(cf,2*ones(size(psif,1),1),'bx')
hold off

If the CWT layer you created does not support your frequency limits, try increasing the number of voices per octave in the layer. For more information, see Practical Introduction to Time-Frequency Analysis Using the Continuous Wavelet Transform and Continuous and Discrete Wavelet Transforms.

Reset Frequency Limits to Default Values in Deep Network Designer

This example shows how to reset the frequency limits to default values after changing the wavelet of a cwtLayer in Deep Network Designer.

Suppose you are editing a deep learning network using Deep Network Designer. The network has a cwtLayer that uses the Morse wavelet and has frequency limits set at [0.1,0.3].

Use the drop-down list to change the Wavelet to amor. The frequency limits do not change, but the number of weights does change.

To change the frequency limits to default values appropriate for the amor wavelet, first do one of the following:

• Delete the FrequencyLimits 0.1,0.3.

• Set the FrequencyLimits to [].

Then click the mouse outside the FrequencyLimits edit field to move the focus. The app automatically populates FrequencyLimits with default values for the amor wavelet. The number of weights also changes to reflect the new limits.

Tip: If you change the frequency limits, and later want to restore the default values, follow the same steps.

More About

Layer Output Format

cwtLayer formats the output as "SCBT", a sequence of 1-D images where the image height corresponds to scale, or frequency. The second dimension corresponds to the channel, the third dimension corresponds to the batch, and the fourth dimension corresponds to time.

• You can feed the output of cwtLayer unchanged to a 1-D convolutional layer when you want to convolve along the frequency ("S") dimension. For more information, see convolution1dLayer (Deep Learning Toolbox).

• To feed the output of cwtLayer to a 1-D convolutional layer when you want to convolve along the time ("T") dimension, you must place a flatten layer after the cwtLayer. For more information, see flattenLayer (Deep Learning Toolbox).

• You can feed the output of cwtLayer unchanged to a 2-D convolutional layer when you want to convolve along the frequency ("S") and time ("T") dimensions jointly. For more information, see convolution2dLayer (Deep Learning Toolbox).

• To use cwtLayer as part of a recurrent neural network, you must place a flatten layer after the cwtLayer. For more information, see lstmLayer (Deep Learning Toolbox) and gruLayer (Deep Learning Toolbox).

• To use the output of cwtLayer with a fully connected layer as part of a classification workflow, you must reduce the time ("T") dimension of the output so that it has size 1. To reduce the time dimension of the output, place a global pooling layer before the fully connected layer. For more information, see globalAveragePooling2dLayer (Deep Learning Toolbox) and fullyConnectedLayer (Deep Learning Toolbox).

Version History

Introduced in R2022b