Operators concatenation

This example shows how to use ‘stacking’ operators such as pylops.VStack, pylops.HStack, pylops.Block, pylops.BlockDiag, and pylops.Kronecker.

These operators allow for different combinations of multiple linear operators in a single operator. Such functionalities are used within PyLops as the basis for the creation of complex operators as well as in the definition of various types of optimization problems with regularization or preconditioning.

import numpy as np
import matplotlib.pyplot as plt

import pylops

plt.close('all')

Let’s start by defining two second derivatives pylops.SecondDerivative that we will be using in this example.

D2hop = pylops.SecondDerivative(11 * 21, dims=[11, 21], dir=1, dtype='float32')
D2vop = pylops.SecondDerivative(11 * 21, dims=[11, 21], dir=0, dtype='float32')

Chaining of operators represents the simplest concatenation that can be performed between two or more linear operators. This can be easily achieved using the * symbol

\[\mathbf{D_{cat}}= \mathbf{D_v} \mathbf{D_h}\]
Nv, Nh = 11, 21
X = np.zeros((Nv, Nh))
X[int(Nv/2), int(Nh/2)] = 1

D2op = D2vop*D2hop
Y = np.reshape(D2op*X.flatten(), (Nv, Nh))

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle('Chain', fontsize=14,
             fontweight='bold', y=0.95)
im = axs[0].imshow(X, interpolation='nearest')
axs[0].axis('tight')
axs[0].set_title(r'$x$')
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation='nearest')
axs[1].axis('tight')
axs[1].set_title(r'$y=(D_x+D_y) x$')
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)
../_images/sphx_glr_plot_stacking_001.png

We now want to vertically stack three operators

\[\begin{split}\mathbf{D_{Vstack}} = \begin{bmatrix} \mathbf{D_v} \\ \mathbf{D_h} \end{bmatrix}, \qquad \mathbf{y} = \begin{bmatrix} \mathbf{D_v}\mathbf{x} \\ \mathbf{D_h}\mathbf{x} \end{bmatrix}\end{split}\]
Nv, Nh = 11, 21
X = np.zeros((Nv, Nh))
X[int(Nv/2), int(Nh/2)] = 1
Dstack = pylops.VStack([D2vop, D2hop])

Y = np.reshape(Dstack * X.flatten(), (Nv * 2, Nh))

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle('Vertical stacking', fontsize=14,
             fontweight='bold', y=0.95)
im = axs[0].imshow(X, interpolation='nearest')
axs[0].axis('tight')
axs[0].set_title(r'$x$')
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation='nearest')
axs[1].axis('tight')
axs[1].set_title(r'$y$')
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)
../_images/sphx_glr_plot_stacking_002.png

Similarly we can now horizontally stack three operators

\[\mathbf{D_{Hstack}} = \begin{bmatrix} \mathbf{D_v} & 0.5*\mathbf{D_v} & -1*\mathbf{D_h} \end{bmatrix}, \qquad \mathbf{y} = \mathbf{D_v}\mathbf{x}_1 + 0.5*\mathbf{D_v}\mathbf{x}_2 - \mathbf{D_h}\mathbf{x}_3\]
Nv, Nh = 11, 21
X = np.zeros((Nv*3, Nh))
X[int(Nv/2), int(Nh/2)] = 1
X[int(Nv/2) + Nv, int(Nh/2)] = 1
X[int(Nv/2) + 2*Nv, int(Nh/2)] = 1

Hstackop = pylops.HStack([D2vop, 0.5 * D2vop, -1 * D2hop])
Y = np.reshape(Hstackop*X.flatten(), (Nv, Nh))

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle('Horizontal stacking', fontsize=14,
             fontweight='bold', y=0.95)
im = axs[0].imshow(X, interpolation='nearest')
axs[0].axis('tight')
axs[0].set_title(r'$x$')
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation='nearest')
axs[1].axis('tight')
axs[1].set_title(r'$y$')
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)
../_images/sphx_glr_plot_stacking_003.png

We can even stack them both horizontally and vertically such that we create a block operator

\[\begin{split}\mathbf{D_{Block}} = \begin{bmatrix} \mathbf{D_v} & 0.5*\mathbf{D_v} & -1*\mathbf{D_h} \\ \mathbf{D_h} & 2*\mathbf{D_h} & \mathbf{D_v} \\ \end{bmatrix}, \qquad \mathbf{y} = \begin{bmatrix} \mathbf{D_v} \mathbf{x_1} + 0.5*\mathbf{D_v} \mathbf{x_2} - \mathbf{D_h} \mathbf{x_3} \\ \mathbf{D_h} \mathbf{x_1} + 2*\mathbf{D_h} \mathbf{x_2} + \mathbf{D_v} \mathbf{x_3} \end{bmatrix}\end{split}\]
Bop = pylops.Block([[D2vop, 0.5 * D2vop, -1 * D2hop],
                    [D2hop, 2 * D2hop, D2vop]])
Y = np.reshape(Bop*X.flatten(), (2*Nv, Nh))

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle('Block', fontsize=14,
             fontweight='bold', y=0.95)
im = axs[0].imshow(X, interpolation='nearest')
axs[0].axis('tight')
axs[0].set_title(r'$x$')
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation='nearest')
axs[1].axis('tight')
axs[1].set_title(r'$y$')
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)
../_images/sphx_glr_plot_stacking_004.png

Finally we can use the block-diagonal operator to apply three operators to three different subset of the model and data

\[\begin{split}\mathbf{D_{BDiag}} = \begin{bmatrix} \mathbf{D_v} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & 0.5*\mathbf{D_v} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & -\mathbf{D_h} \end{bmatrix}, \qquad \mathbf{y} = \begin{bmatrix} \mathbf{D_v} \mathbf{x_1} \\ 0.5*\mathbf{D_v} \mathbf{x_2} \\ -\mathbf{D_h} \mathbf{x_3} \end{bmatrix}\end{split}\]
BD = pylops.BlockDiag([D2vop, 0.5 * D2vop, -1 * D2hop])
Y = np.reshape(BD*np.ndarray.flatten(X), (11*3, 21))

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle('Block-diagonal', fontsize=14,
             fontweight='bold', y=0.95)
im = axs[0].imshow(X, interpolation='nearest')
axs[0].axis('tight')
axs[0].set_title(r'$x$')
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation='nearest')
axs[1].axis('tight')
axs[1].set_title(r'$y$')
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)
../_images/sphx_glr_plot_stacking_005.png

Finally we use the Kronecker operator and replicate this example on wiki.

\[\begin{split}\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \otimes \begin{bmatrix} 0 & 5 \\ 6 & 7 \\ \end{bmatrix} = \begin{bmatrix} 0 & 5 & 0 & 10 \\ 6 & 7 & 12 & 14 \\ 0 & 15 & 0 & 20 \\ 18 & 21 & 24 & 28 \\ \end{bmatrix}\end{split}\]
A = np.array([[1, 2], [3, 4]])
B = np.array([[0, 5], [6, 7]])
AB = np.kron(A, B)

n1, m1 = A.shape
n2, m2 = B.shape

Aop = pylops.MatrixMult(A)
Bop = pylops.MatrixMult(B)

ABop = pylops.Kronecker(Aop, Bop)
x = np.ones(m1*m2)

y = AB.dot(x)
yop = ABop*x
xinv = ABop / yop

print('AB = \n', AB)

print('x = ', x)
print('y = ', y)
print('yop = ', yop)
print('xinv = ', x)

Out:

AB =
 [[ 0  5  0 10]
 [ 6  7 12 14]
 [ 0 15  0 20]
 [18 21 24 28]]
x =  [1. 1. 1. 1.]
y =  [15. 39. 35. 91.]
yop =  [15. 39. 35. 91.]
xinv =  [1. 1. 1. 1.]

We can also use pylops.Kronecker to do something more interesting. Any operator can in fact be applied on a single direction of a multi-dimensional input array if combined with an pylops.Identity operator via Kronecker product. We apply here the pylops.FirstDerivative to the second dimension of the model.

Note that for those operators whose implementation allows their application to a single axis via the dir parameter, using the Kronecker product would lead to slower performance. Nevertheless, the Kronecker product allows any other operator to be applied to a single dimension.

Nv, Nh = 11, 21

Iop = pylops.Identity(Nv, dtype='float32')
D2hop = pylops.FirstDerivative(Nh, dtype='float32')

X = np.zeros((Nv, Nh))
X[Nv//2, Nh//2] = 1
D2hop = pylops.Kronecker(Iop, D2hop)

Y = D2hop*X.ravel()
Y = Y.reshape(Nv, Nh)

fig, axs = plt.subplots(1, 2, figsize=(10, 3))
fig.suptitle('Kronecker', fontsize=14,
             fontweight='bold', y=0.95)
im = axs[0].imshow(X, interpolation='nearest')
axs[0].axis('tight')
axs[0].set_title(r'$x$')
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(Y, interpolation='nearest')
axs[1].axis('tight')
axs[1].set_title(r'$y$')
plt.colorbar(im, ax=axs[1])
plt.tight_layout()
plt.subplots_adjust(top=0.8)
../_images/sphx_glr_plot_stacking_006.png

Total running time of the script: ( 0 minutes 2.506 seconds)

Gallery generated by Sphinx-Gallery