Identity

This example shows how to use the pylops.Identity operator to transfer model into data and viceversa.

import torch
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.gridspec as pltgs

import pylops_gpu

plt.close('all')

Let’s define an identity operator \(\mathbf{I}\) with same number of elements for data \(N\) and model \(M\).

N, M = 5, 5
x = torch.arange(M, dtype=torch.int)
Iop = pylops_gpu.Identity(M, dtype=torch.int)

y = Iop * x
xadj = Iop.H * y

gs = pltgs.GridSpec(1, 6)
fig = plt.figure(figsize=(7, 3))
ax = plt.subplot(gs[0, 0:3])
im = ax.imshow(np.eye(N), cmap='rainbow')
ax.set_title('A', size=20, fontweight='bold')
ax.set_xticks(np.arange(N-1)+0.5)
ax.set_yticks(np.arange(M-1)+0.5)
ax.grid(linewidth=3, color='white')
ax.xaxis.set_ticklabels([])
ax.yaxis.set_ticklabels([])
ax = plt.subplot(gs[0, 3])
ax.imshow(x[:, np.newaxis], cmap='rainbow')
ax.set_title('x', size=20, fontweight='bold')
ax.set_xticks([])
ax.set_yticks(np.arange(M-1)+0.5)
ax.grid(linewidth=3, color='white')
ax.xaxis.set_ticklabels([])
ax.yaxis.set_ticklabels([])
ax = plt.subplot(gs[0, 4])
ax.text(0.35, 0.5, '=', horizontalalignment='center',
        verticalalignment='center', size=40, fontweight='bold')
ax.axis('off')
ax = plt.subplot(gs[0, 5])
ax.imshow(y[:, np.newaxis], cmap='rainbow')
ax.set_title('y', size=20, fontweight='bold')
ax.set_xticks([])
ax.set_yticks(np.arange(N - 1) + 0.5)
ax.grid(linewidth=3, color='white')
ax.xaxis.set_ticklabels([])
ax.yaxis.set_ticklabels([])
fig.colorbar(im, ax=ax, ticks=[0, 1], pad=0.3, shrink=0.7)

Out:

<matplotlib.colorbar.Colorbar object at 0x7fbc6ba4ffd0>

Similarly we can consider the case with data bigger than model

N, M = 10, 5
x = torch.arange(M, dtype=torch.int)
Iop = pylops_gpu.Identity(N, M, dtype=torch.int)

y = Iop*x
xadj = Iop.H*y

print('x = %s ' % x)
print('I*x = %s ' % y)
print('I\'*y = %s ' % xadj)

Out:

x = tensor([0, 1, 2, 3, 4], dtype=torch.int32)
I*x = tensor([0, 1, 2, 3, 4, 0, 0, 0, 0, 0], dtype=torch.int32)
I'*y = tensor([0, 1, 2, 3, 4], dtype=torch.int32)

and model bigger than data

N, M = 5, 10
x = torch.arange(M, dtype=torch.int)
Iop = pylops_gpu.Identity(N, M, dtype=torch.int)

y = Iop * x
xadj = Iop.H * y

print('x = %s ' % x)
print('I*x = %s ' % y)
print('I\'*y = %s ' % xadj)

Out:

x = tensor([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], dtype=torch.int32)
I*x = tensor([0, 1, 2, 3, 4], dtype=torch.int32)
I'*y = tensor([0, 1, 2, 3, 4, 0, 0, 0, 0, 0], dtype=torch.int32)