spaths = paths(sf)
returns the scattering paths for all orders of the scattering framework,
sf. spaths is a cell array of MATLAB^{®} tables with n elements, where n is the
number of orders in the scattering framework.

[spaths,npaths] = paths(sf)
returns the number of paths in each order as n-by-1 column vector, where
n is the number of orders in the scattering framework. The sum of the
elements of npaths is the total number of scattering paths.

This example compares the number of paths in a wavelet image scattering frameworks with three orders.

Create an image scattering framework with an image size of 256-by-256 and invariance scale equal to the minimum of the image size. The default OptimizePath value is 1 (true).

sf = waveletScattering2('ImageSize',[256 256],'InvarianceScale',128)

Obtain the number of scattering paths in each order. Display the total number of scattering paths.

[spaths,npaths] = paths(sf);
sum(npaths)

ans = 391

Set the OptimizePath value of the framework to false. Display the total number of scattering paths. For this framework, the scattering transform does not reduce the number of paths to compute based on a bandwidth consideration.

This example shows how the OptimizePath property can affect the scattering paths that include a specific wavelet.

Create the default wavelet image scattering framework. Obtain all the wavelet filters and center spatial frequencies for the framework. Obtain all the framework scattering paths. Display the total number of paths.

sf = waveletScattering2
[~,psifilters,f] = filterbank(sf);
[spaths,npaths] = paths(sf);
disp(['Total Number of Paths: ',num2str(sum(npaths))])

sf =
waveletScattering2 with properties:
ImageSize: [128 128]
InvarianceScale: 64
NumRotations: [6 6]
QualityFactors: [1 1]
Precision: 'single'
OversamplingFactor: 0
OptimizePath: 1
Total Number of Paths: 241

Display the number of wavelet filters in each filter bank.

disp(['Filter Bank 1: ',num2str(size(psifilters{1},3))]);
disp(['Filter Bank 2: ',num2str(size(psifilters{2},3))]);

Filter Bank 1: 24
Filter Bank 2: 24

Choose a wavelet from the first filter bank and display its spatial center frequency. Use spaths to find all the three-element paths that include the chosen wavelet. Display the paths.

waveletA = 14;
disp(['Center Frequency: ',num2str(f{1}(waveletA,:))]);
ind = find(spaths{3}.path(:,2)==waveletA);
spaths{3}(ind,:)

Plot the center frequencies of the wavelet filters on the paths.

plot(f{1}(waveletA,1),f{1}(waveletA,2),'k^');
xlabel('f_x')
ylabel('f_y')
hold on
waveletBs = spaths{3}.path(ind,3);
plot(f{2}(waveletBs,1),f{2}(waveletBs,2),'bx');
grid on
legend('First Filter Bank Wavelet','Second Filter Bank Wavelets',...'Location','northeastoutside')

Now set the OptimizePath property of the scattering framework sf to false. Obtain the wavelet filters, center spatial frequencies, and scattering paths of the framework.

sf.OptimizePath = false
[~,psifilters2,f2] = filterbank(sf);
[spaths2,npaths2] = paths(sf);
disp(['Total Number of Paths: ',num2str(sum(npaths2))])

sf =
waveletScattering2 with properties:
ImageSize: [128 128]
InvarianceScale: 64
NumRotations: [6 6]
QualityFactors: [1 1]
Precision: 'single'
OversamplingFactor: 0
OptimizePath: 0
Total Number of Paths: 385

Choose the same wavelet as above. To confirm it is the same wavelet, display its spatial center frequency. Use spaths to find all the three-element paths that include the wavelet. Because OptimizePath is set to false, the wavelet filter has more children.

waveletA = 14;
disp(['Center Frequency: ',num2str(f2{1}(waveletA,:))]);
ind = find(spaths2{3}.path(:,2)==waveletA);
spaths2{3}(ind,:)

Plot the center frequencies of the wavelet filters on the paths. Some of child filters have center frequencies higher than the chosen wavelet.

figure
plot(f2{1}(waveletA,1),f2{1}(waveletA,2),'k^');
xlabel('f_x')
ylabel('f_y')
hold on
waveletBs = spaths2{3}.path(ind,3);
plot(f2{2}(waveletBs,1),f2{2}(waveletBs,2),'bx');
grid on
legend('First Filter Bank Wavelet','Second Filter Bank Wavelets',...'Location','northeastoutside')

Scattering paths of all orders of the scattering framework, returned as a cell array
of MATLAB tables. spaths has n elements,
where n is the number of orders in the scattering framework.

Each MATLAB table in spaths contains a single variable,
path. The variable path is a row vector with one
column for each element of the path. The scalar 0 denotes the original image. Positive
integers in the Lth column denote the corresponding wavelet filter in
the (L−1)th filter bank. Wavelet bandpass filters are ordered by
decreasing center frequency. There are NumRotations wavelets per
center frequency pair.

npaths — Number of scattering paths column vector

Number of scattering paths in each order of the scattering framework.
npaths is a no-by-1 column vector where
no is the number of orders in the scattering framework. The sum of
the elements of npaths is the total number of scattering
paths.

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