Scattering disk display

This script reproduces concentric circles that encode Scattering coefficient’s energy as described in “Invariant Scattering Convolution Networks” by Bruna and Mallat. Here, for the sake of simplicity, we only consider first order scattering.

Author: Edited by: Edouard Oyallon

import matplotlib as mpl
import as cm
import matplotlib.pyplot as plt
import numpy as np
from kymatio import Scattering2D
from PIL import Image
import os

img_name = os.path.join(os.getcwd(),"images/digit.png")

Scattering computations

First, we read the input digit:

src_img ='L').resize((32,32))
src_img = np.array(src_img)
print("img shape: ", src_img.shape)


img shape:  (32, 32)

We compute a Scattering Transform with L=6 angles and J=3 scales. Rotating a wavelet \psi by \pi is equivalent to consider its conjugate in fourier: \hat\psi_{\pi}(\omega)=\hat\psi(r_{-\pi}\omega)^*.

Combining this and the fact that a real signal has a Hermitian symmetry implies that it is usually sufficient to use the angles \{\frac{\pi l}{L}\}_{l\leq L} at computation time. For consistency, we will however display \{\frac{2\pi l}{L}\}_{l\leq 2L}, which implies that our visualization will be redundant and have a symmetry by rotation of \pi.

L = 6
J = 3
scattering = Scattering2D(J=J, shape=src_img.shape, L=L, max_order=1, frontend='numpy')

We now compute the scattering coefficients:

src_img_tensor = src_img.astype(np.float32) / 255.

scattering_coefficients = scattering(src_img_tensor)
print("coeffs shape: ", scattering_coefficients.shape)
# Invert colors
scattering_coefficients = -scattering_coefficients


coeffs shape:  (19, 4, 4)

We skip the low pass filter…

scattering_coefficients = scattering_coefficients[1:, :, :]
norm = mpl.colors.Normalize(scattering_coefficients.min(), scattering_coefficients.max(), clip=True)
mapper = cm.ScalarMappable(norm=norm, cmap="gray")
nb_coeffs, window_rows, window_columns = scattering_coefficients.shape

Figure reproduction

Now we can reproduce a figure that displays the energy of the first order Scattering coefficient, which are given by \{\mid x\star\psi_{j,\theta}\mid\star\phi_J|\}_{j,\theta} . Here, each scattering coefficient is represented on the polar plane. The polar radius and angle correspond respectively to the scale j and the rotation \theta applied to the mother wavelet.

Observe that as predicted, the visualization exhibit a redundancy and a symmetry.

fig,ax = plt.subplots()

plt.imshow(1-src_img,cmap='gray',interpolation='nearest', aspect='auto')
offset = 0.1
for row in range(window_rows):
    for column in range(window_columns):
        ax=fig.add_subplot(window_rows, window_columns, 1 + column + row * window_rows, projection='polar')
        ax.set_ylim(0, 1)
        ax.set_yticklabels([])  # turn off radial tick labels (yticks)
        ax.set_xticklabels([])  # turn off degrees
        # ax.set_theta_zero_location('N')  # 0° to North
        coefficients = scattering_coefficients[:, row, column]
        for j in range(J):
            for l in range(L):
                coeff = coefficients[l + (J - 1 - j) * L]
                color = mpl.colors.to_hex(mapper.to_rgba(coeff))
       *  np.pi / L,
                       height=2*(2**(j-1) / 2**J),
                       width=2 * np.pi / L,
                       bottom=offset + (2**j / 2**J) ,
       * np.pi / L,
                       height=2*(2**(j-1) / 2**J),
                       width=2 * np.pi / L,
                       bottom=offset + (2**j / 2**J) ,

Total running time of the script: ( 0 minutes 1.566 seconds)

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