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plot_stacking.py
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plot_stacking.py
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"""
Operators concatenation
=======================
This example shows how to use 'stacking' operators such as
:py:class:`pylops.VStack`, :py:class:`pylops.HStack`,
:py:class:`pylops.Block`, :py:class:`pylops.BlockDiag`,
and :py:class:`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 :py:class:`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
#
# .. math::
# \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
#
# .. math::
# \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}
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
#
# .. math::
# \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
#
# .. math::
# \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}
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
#
# .. math::
# \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}
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 <https://en.wikipedia.org/wiki/Kronecker_product>`_.
#
# .. math::
# \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}
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)
###############################################################################
# We can also use :py:class:`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 :py:class:`pylops.Identity`
# operator via Kronecker product. We apply here the
# :py:class:`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)