returns the Littlewood-Paley sum for the 2-D filter banks in the 2-D wavelet scattering
lpsum = littlewoodPaleySum(
lpsum is an
M-by-N-by-Nfb matrix, where
M-by-N is the matrix size of the padded filters and
Nfb is the number of filter banks.
Since the scattering transform is contractive, the Littlewood-Paley sums do not exceed 1.
Littlewood-Paley Sum of Image Scattering Network
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,f] = littlewoodPaleySum(sf); max(max(lpsum(:,:,1)))
ans = 1.0000
ans = 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.
f(f>1/2) = f(f>1/2)-1; surf(fftshift(f(:,1)),fftshift(f(:,2)),fftshift(lpsum(:,:,2))) shading interp view(0,90) xlabel('f_x') ylabel('f_y') colorbar title('Q=1')
sf — Wavelet image scattering network
Wavelet image scattering network, specified as a
fb — Filter bank index
positive integer | vector of positive integers
Filter bank index in the image scattering network, specified as a positive integer
or vector of positive integers between 1 and
numfilterbanks( inclusive. The number of
filter banks in
sf is equal to the number of specified
lpsum — Littlewood-Paley sum
real-valued 3-D matrix
Littlewood-Paley sum for the filter banks in the image scattering network
sf, returned as a real-valued 3-D matrix.
lpsum is an
M-by-N-by-L matrix, where
M-by-N is the matrix size of the padded filters
and L does not exceed the number of filter banks in
f — Frequencies
real-valued two-column matrix
Frequencies for the Littlewood-Paley sum, returned as a real-valued two-column
matrix. Frequencies are in cycles per pixel. The first column of
contains the spatial frequencies in the x-direction, and the second
column contains the spatial frequencies in the y-direction. In this
convention, the Fourier transform is 1-periodic in both Fourier variables.