Note
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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)
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)
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)
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)
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)
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)
Total running time of the script: ( 0 minutes 2.253 seconds)