Diagonal

This example shows how to use the pylops.Diagonal operator to perform Element-wise multiplication between the input vector and a vector \(\mathbf{d}\).

In other words, the operator acts as a diagonal operator \(\mathbf{D}\) whose elements along the diagonal are the elements of the vector \(\mathbf{d}\).

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

import pylops

plt.close("all")

Let’s define a diagonal operator \(\mathbf{d}\) with increasing numbers from 0 to N and a unitary model \(\mathbf{x}\).

N = 10
d = np.arange(N)
x = np.ones(N)

Dop = pylops.Diagonal(d)

y = Dop * x
y1 = Dop.H * x

gs = pltgs.GridSpec(1, 6)
fig = plt.figure(figsize=(7, 3))
ax = plt.subplot(gs[0, 0:3])
im = ax.imshow(Dop.matrix(), cmap="rainbow", vmin=0, vmax=N)
ax.set_title("A", size=20, fontweight="bold")
ax.set_xticks(np.arange(N - 1) + 0.5)
ax.set_yticks(np.arange(N - 1) + 0.5)
ax.grid(linewidth=3, color="white")
ax.xaxis.set_ticklabels([])
ax.yaxis.set_ticklabels([])
ax.axis("tight")
ax = plt.subplot(gs[0, 3])
ax.imshow(x[:, np.newaxis], cmap="rainbow", vmin=0, vmax=N)
ax.set_title("x", 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([])
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", vmin=0, vmax=N)
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, N], pad=0.3, shrink=0.7)

Out:

<matplotlib.colorbar.Colorbar object at 0x7f90d1729438>

Similarly we can consider the input model as composed of two or more dimensions. In this case the diagonal operator can be still applied to each element or broadcasted along a specific direction. Let’s start with the simplest case where each element is multipled by a different value

nx, ny = 3, 5
x = np.ones((nx, ny))
print("x =\n%s" % x)

d = np.arange(nx * ny).reshape(nx, ny)
Dop = pylops.Diagonal(d)

y = Dop * x.ravel()
y1 = Dop.H * x.ravel()

print("y = D*x =\n%s" % y.reshape(nx, ny))
print("xadj = D'*x =\n%s " % y1.reshape(nx, ny))

Out:

x =
[[1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1.]]
y = D*x =
[[ 0.  1.  2.  3.  4.]
 [ 5.  6.  7.  8.  9.]
 [10. 11. 12. 13. 14.]]
xadj = D'*x =
[[ 0.  1.  2.  3.  4.]
 [ 5.  6.  7.  8.  9.]
 [10. 11. 12. 13. 14.]]

And we now broadcast

nx, ny = 3, 5
x = np.ones((nx, ny))
print("x =\n%s" % x)

# 1st dim
d = np.arange(nx)
Dop = pylops.Diagonal(d, dims=(nx, ny), dir=0)

y = Dop * x.ravel()
y1 = Dop.H * x.ravel()

print("1st dim: y = D*x =\n%s" % y.reshape(nx, ny))
print("1st dim: xadj = D'*x =\n%s " % y1.reshape(nx, ny))

# 2nd dim
d = np.arange(ny)
Dop = pylops.Diagonal(d, dims=(nx, ny), dir=1)

y = Dop * x.ravel()
y1 = Dop.H * x.ravel()

print("2nd dim: y = D*x =\n%s" % y.reshape(nx, ny))
print("2nd dim: xadj = D'*x =\n%s " % y1.reshape(nx, ny))

Out:

x =
[[1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1.]]
1st dim: y = D*x =
[[0. 0. 0. 0. 0.]
 [1. 1. 1. 1. 1.]
 [2. 2. 2. 2. 2.]]
1st dim: xadj = D'*x =
[[0. 0. 0. 0. 0.]
 [1. 1. 1. 1. 1.]
 [2. 2. 2. 2. 2.]]
2nd dim: y = D*x =
[[0. 1. 2. 3. 4.]
 [0. 1. 2. 3. 4.]
 [0. 1. 2. 3. 4.]]
2nd dim: xadj = D'*x =
[[0. 1. 2. 3. 4.]
 [0. 1. 2. 3. 4.]
 [0. 1. 2. 3. 4.]]