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littlewoodPaleySum

Littlewood-Paley sum for 2-D wavelet scattering network

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Syntax

lpsum = littlewoodPaleySum(sf)
lpsum = littlewoodPaleySum(sf,fb)
[lpsum,fx,fy] = littlewoodPaleySum(___)

Description

lpsum = littlewoodPaleySum(sf) returns the Littlewood-Paley sum for the 2-D filter banks in the 2-D wavelet scattering network sf.

Because the scattering transform is contractive, the Littlewood-Paley sums do not exceed 1.

lpsum = littlewoodPaleySum(sf,fb) returns the Littlewood-Paley sum for the specified filter banks fb.

[lpsum,fx,fy] = littlewoodPaleySum(___) returns the spatial frequencies in the x- and y-directions for the Littlewood-Paley sum.

example

Examples

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Open Live Script

This example shows how to obtain and display the Littlewood-Paley sum of an image scattering network.

Create a scattering network with two filter banks and quality factors of 2 and 1, respectively.

sf = waveletScattering2(QualityFactors=[2 1]);

Obtain the Littlewood-Paley sums and spatial frequencies of the two filter banks. Display the maximum value of the sums. Since the scattering transform is contractive, the sums do not exceed 1.

[lpsum,fx,fy] = littlewoodPaleySum(sf);
max(max(lpsum(:,:,1)))
ans = single

1.0000

max(max(lpsum(:,:,2)))
ans = single

1.0000

Display the Littlewood-Paley sum of the second filter bank with the zero frequency centered. Note the 2-D Morlet filter bank used in the scattering transform is not designed to capture the highest spatial frequencies jointly in the x- and y-directions.

fx(fx>1/2) = fx(fx>1/2)-1;
fy(fy>1/2) = fy(fy>1/2)-1;
surf(fftshift(fx),fftshift(fy),fftshift(lpsum(:,:,2)))
shading interp
view(0,90)
xlabel("f_x")
ylabel("f_y")
colorbar
title("Q=1")

Figure contains an axes object. The axes object with title Q=1, xlabel f indexOf x baseline f_x, ylabel f indexOf y baseline f_y contains an object of type surface.

Input Arguments

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Wavelet image scattering network, specified as a waveletScattering2 object.

Filter bank indices in the image scattering network, specified as a positive integer or vector of positive integers between 1 and numfilterbanks(sf) inclusive. The number of filter banks in sf is equal to the number of specified QualityFactors in sf.

Output Arguments

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Littlewood-Paley sum for the filter banks in the image scattering network sf, returned as a real-valued 3-D array. lpsum is an M-by-N-by-L array, where M-by-N is the matrix size of the padded filters and L does not exceed the number of filter banks in sf. If you specify indices fb, L is the number of unique elements in fb. Otherwise, L is the number of filter banks.

Since R2024a

Spatial frequencies for the Littlewood-Paley sum, returned as a pair of real-valued vectors. fx and fy are the spatial frequencies in the x- and y- dimensions, respectively. Frequencies are in cycles per pixel. In this convention, the Fourier transform is 1-periodic in both Fourier variables.

Version History

Introduced in R2019a

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See Also