<reponame>SoftwareUnderstanding/inspect4py
r"""
Radon Transform
===============
This example shows how to use the :py:class:`pylops.signalprocessing.Radon2D`
and :py:class:`pylops.signalprocessing.Radon3D` operators to apply the Radon
Transform to 2-dimensional or 3-dimensional signals, respectively.
In our implementation both linear, parabolic and hyperbolic parametrization
can be chosen.
"""
import numpy as np
import matplotlib.pyplot as plt

import pylops

plt.close('all')

###############################################################################
# Let's start by creating an empty 2d matrix of size :math:`n_{p_x} \times n_t`
# and add a single spike in it. We will see that applying the forward
# Radon operator will result in a single event (linear, parabolic or
# hyperbolic) in the resulting data vector.
nt, nh = 41, 51
npx, pxmax = 41, 1e-2

dt, dh = 0.005, 1
t = np.arange(nt) * dt
h = np.arange(nh) * dh
px = np.linspace(0, pxmax, npx)

x = np.zeros((npx, nt))
x[4, nt//2] = 1

###############################################################################
# We can now define our operators for different parametric curves and apply
# them to the input model vector. We also apply the adjoint to the resulting
# data vector.
RLop = pylops.signalprocessing.Radon2D(t, h, px, centeredh=True,
                                       kind='linear', interp=False,
                                       engine='numpy')
RPop = pylops.signalprocessing.Radon2D(t, h, px, centeredh=True,
                                       kind='parabolic', interp=False,
                                       engine='numpy')
RHop = pylops.signalprocessing.Radon2D(t, h, px, centeredh=True,
                                       kind='hyperbolic', interp=False,
                                       engine='numpy')

# forward
yL = RLop * x.ravel()
yP = RPop * x.ravel()
yH = RHop * x.ravel()
yL = yL.reshape(nh, nt)
yP = yP.reshape(nh, nt)
yH = yH.reshape(nh, nt)

# adjoint
xadjL = RLop.H * yL.ravel()
xadjP = RPop.H * yP.ravel()
xadjH = RHop.H * yH.ravel()
xadjL = xadjL.reshape(npx, nt)
xadjP = xadjP.reshape(npx, nt)
xadjH = xadjH.reshape(npx, nt)

###############################################################################
# Let's now visualize the input model in the Radon domain, the data, and
# the adjoint model the different parametric curves.

fig, axs = plt.subplots(2, 4, figsize=(10, 6))
axs[0][0].imshow(x.T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[0][0].set_title('Input model')
axs[0][0].axis('tight')
axs[0][1].imshow(yL.T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(h[0], h[-1], t[-1], t[0]))
axs[0][1].set_title('Linear data')
axs[0][1].axis('tight')
axs[0][2].imshow(yP.T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(h[0], h[-1], t[-1], t[0]))
axs[0][2].set_title('Parabolic data')
axs[0][2].axis('tight')
axs[0][3].imshow(yH.T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(h[0], h[-1], t[-1], t[0]))
axs[0][3].set_title('Hyperbolic data')
axs[0][3].axis('tight')
axs[1][1].imshow(xadjL.T, vmin=-20, vmax=20, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][0].axis('off')
axs[1][1].set_title('Linear adjoint')
axs[1][1].axis('tight')
axs[1][2].imshow(xadjP.T, vmin=-20, vmax=20, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][2].set_title('Parabolic adjoint')
axs[1][2].axis('tight')
axs[1][3].imshow(xadjH.T, vmin=-20, vmax=20, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][3].set_title('Hyperbolic adjoint')
axs[1][3].axis('tight')
fig.tight_layout()

###############################################################################
# As we can see in the bottom figures, the adjoint Radon transform is far
# from being close to the inverse Radon transform, i.e.
# :math:`\mathbf{R^H}\mathbf{R} \neq \mathbf{I}` (compared to the case of FFT
# where the adjoint and inverse are equivalent, i.e.
# :math:`\mathbf{F^H}\mathbf{F} = \mathbf{I}`). In fact when we apply the
# adjoint Radon Transform we obtain a *model* that
# is a smoothed version of the original model polluted by smearing and
# artifacts. In tutorial :ref:`sphx_glr_tutorials_radonfiltering.py` we will
# exploit a sparsity-promiting Radon transform to perform filtering of unwanted
# signals from an input data.
#
# Finally we repeat the same exercise with 3d data.
nt, ny, nx = 21, 21, 11
npy, pymax = 13, 5e-3
npx, pxmax = 11, 5e-3

dt, dy, dx = 0.005, 1, 1
t = np.arange(nt) * dt
hy = np.arange(ny) * dy
hx = np.arange(nx) * dx

py = np.linspace(0, pymax, npy)
px = np.linspace(0, pxmax, npx)

x = np.zeros((npy, npx, nt))
x[npy//2, npx//2-2, nt//2] = 1

RLop = pylops.signalprocessing.Radon3D(t, hy, hx, py, px, centeredh=True,
                                       kind='linear', interp=False,
                                       engine='numpy')
RPop = pylops.signalprocessing.Radon3D(t, hy, hx, py, px, centeredh=True,
                                       kind='parabolic', interp=False,
                                       engine='numpy')
RHop = pylops.signalprocessing.Radon3D(t, hy, hx, py, px, centeredh=True,
                                       kind='hyperbolic', interp=False,
                                       engine='numpy')

# forward
yL = RLop * x.ravel()
yP = RPop * x.ravel()
yH = RHop * x.ravel()
yL = yL.reshape(ny, nx, nt)
yP = yP.reshape(ny, nx, nt)
yH = yH.reshape(ny, nx, nt)

# adjoint
xadjL = RLop.H * yL.ravel()
xadjP = RPop.H * yP.ravel()
xadjH = RHop.H * yH.ravel()
xadjL = xadjL.reshape(npy, npx, nt)
xadjP = xadjP.reshape(npy, npx, nt)
xadjH = xadjH.reshape(npy, npx, nt)

# plotting
fig, axs = plt.subplots(2, 4, figsize=(10, 6))
axs[0][0].imshow(x[npy//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[0][0].set_title('Input model')
axs[0][0].axis('tight')
axs[0][1].imshow(yL[ny//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hx[0], hx[-1], t[-1], t[0]))
axs[0][1].set_title('Linear data')
axs[0][1].axis('tight')
axs[0][2].imshow(yP[ny//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hx[0], hx[-1], t[-1], t[0]))
axs[0][2].set_title('Parabolic data')
axs[0][2].axis('tight')
axs[0][3].imshow(yH[ny//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hx[0], hx[-1], t[-1], t[0]))
axs[0][3].set_title('Hyperbolic data')
axs[0][3].axis('tight')
axs[1][1].imshow(xadjL[npy//2].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][0].axis('off')
axs[1][1].set_title('Linear adjoint')
axs[1][1].axis('tight')
axs[1][2].imshow(xadjP[npy//2].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][2].set_title('Parabolic adjoint')
axs[1][2].axis('tight')
axs[1][3].imshow(xadjH[npy//2].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][3].set_title('Hyperbolic adjoint')
axs[1][3].axis('tight')
fig.tight_layout()

fig, axs = plt.subplots(2, 4, figsize=(10, 6))
axs[0][0].imshow(x[:, npx//2-2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(py[0], py[-1], t[-1], t[0]))
axs[0][0].set_title('Input model')
axs[0][0].axis('tight')
axs[0][1].imshow(yL[:, nx//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hy[0], hy[-1], t[-1], t[0]))
axs[0][1].set_title('Linear data')
axs[0][1].axis('tight')
axs[0][2].imshow(yP[:, nx//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hy[0], hy[-1], t[-1], t[0]))
axs[0][2].set_title('Parabolic data')
axs[0][2].axis('tight')
axs[0][3].imshow(yH[:, nx//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hy[0], hy[-1], t[-1], t[0]))
axs[0][3].set_title('Hyperbolic data')
axs[0][3].axis('tight')
axs[1][1].imshow(xadjL[:, npx//2-5].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(py[0], py[-1], t[-1], t[0]))
axs[1][0].axis('off')
axs[1][1].set_title('Linear adjoint')
axs[1][1].axis('tight')
axs[1][2].imshow(xadjP[:, npx//2-2].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(py[0], py[-1], t[-1], t[0]))
axs[1][2].set_title('Parabolic adjoint')
axs[1][2].axis('tight')
axs[1][3].imshow(xadjH[:, npx//2-2].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(py[0], py[-1], t[-1], t[0]))
axs[1][3].set_title('Hyperbolic adjoint')
axs[1][3].axis('tight')
fig.tight_layout()

r"""
Radon Transform
===============
This example shows how to use the :py:class:`pylops.signalprocessing.Radon2D`
and :py:class:`pylops.signalprocessing.Radon3D` operators to apply the Radon
Transform to 2-dimensional or 3-dimensional signals, respectively.
In our implementation both linear, parabolic and hyperbolic parametrization
can be chosen.
"""
import numpy as np
import matplotlib.pyplot as plt

import pylops

plt.close('all')

###############################################################################
# Let's start by creating an empty 2d matrix of size :math:`n_{p_x} \times n_t`
# and add a single spike in it. We will see that applying the forward
# Radon operator will result in a single event (linear, parabolic or
# hyperbolic) in the resulting data vector.
nt, nh = 41, 51
npx, pxmax = 41, 1e-2

dt, dh = 0.005, 1
t = np.arange(nt) * dt
h = np.arange(nh) * dh
px = np.linspace(0, pxmax, npx)

x = np.zeros((npx, nt))
x[4, nt//2] = 1

###############################################################################
# We can now define our operators for different parametric curves and apply
# them to the input model vector. We also apply the adjoint to the resulting
# data vector.
RLop = pylops.signalprocessing.Radon2D(t, h, px, centeredh=True,
                                       kind='linear', interp=False,
                                       engine='numpy')
RPop = pylops.signalprocessing.Radon2D(t, h, px, centeredh=True,
                                       kind='parabolic', interp=False,
                                       engine='numpy')
RHop = pylops.signalprocessing.Radon2D(t, h, px, centeredh=True,
                                       kind='hyperbolic', interp=False,
                                       engine='numpy')

# forward
yL = RLop * x.ravel()
yP = RPop * x.ravel()
yH = RHop * x.ravel()
yL = yL.reshape(nh, nt)
yP = yP.reshape(nh, nt)
yH = yH.reshape(nh, nt)

# adjoint
xadjL = RLop.H * yL.ravel()
xadjP = RPop.H * yP.ravel()
xadjH = RHop.H * yH.ravel()
xadjL = xadjL.reshape(npx, nt)
xadjP = xadjP.reshape(npx, nt)
xadjH = xadjH.reshape(npx, nt)

###############################################################################
# Let's now visualize the input model in the Radon domain, the data, and
# the adjoint model the different parametric curves.

fig, axs = plt.subplots(2, 4, figsize=(10, 6))
axs[0][0].imshow(x.T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[0][0].set_title('Input model')
axs[0][0].axis('tight')
axs[0][1].imshow(yL.T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(h[0], h[-1], t[-1], t[0]))
axs[0][1].set_title('Linear data')
axs[0][1].axis('tight')
axs[0][2].imshow(yP.T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(h[0], h[-1], t[-1], t[0]))
axs[0][2].set_title('Parabolic data')
axs[0][2].axis('tight')
axs[0][3].imshow(yH.T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(h[0], h[-1], t[-1], t[0]))
axs[0][3].set_title('Hyperbolic data')
axs[0][3].axis('tight')
axs[1][1].imshow(xadjL.T, vmin=-20, vmax=20, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][0].axis('off')
axs[1][1].set_title('Linear adjoint')
axs[1][1].axis('tight')
axs[1][2].imshow(xadjP.T, vmin=-20, vmax=20, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][2].set_title('Parabolic adjoint')
axs[1][2].axis('tight')
axs[1][3].imshow(xadjH.T, vmin=-20, vmax=20, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][3].set_title('Hyperbolic adjoint')
axs[1][3].axis('tight')
fig.tight_layout()

###############################################################################
# As we can see in the bottom figures, the adjoint Radon transform is far
# from being close to the inverse Radon transform, i.e.
# :math:`\mathbf{R^H}\mathbf{R} \neq \mathbf{I}` (compared to the case of FFT
# where the adjoint and inverse are equivalent, i.e.
# :math:`\mathbf{F^H}\mathbf{F} = \mathbf{I}`). In fact when we apply the
# adjoint Radon Transform we obtain a *model* that
# is a smoothed version of the original model polluted by smearing and
# artifacts. In tutorial :ref:`sphx_glr_tutorials_radonfiltering.py` we will
# exploit a sparsity-promiting Radon transform to perform filtering of unwanted
# signals from an input data.
#
# Finally we repeat the same exercise with 3d data.
nt, ny, nx = 21, 21, 11
npy, pymax = 13, 5e-3
npx, pxmax = 11, 5e-3

dt, dy, dx = 0.005, 1, 1
t = np.arange(nt) * dt
hy = np.arange(ny) * dy
hx = np.arange(nx) * dx

py = np.linspace(0, pymax, npy)
px = np.linspace(0, pxmax, npx)

x = np.zeros((npy, npx, nt))
x[npy//2, npx//2-2, nt//2] = 1

RLop = pylops.signalprocessing.Radon3D(t, hy, hx, py, px, centeredh=True,
                                       kind='linear', interp=False,
                                       engine='numpy')
RPop = pylops.signalprocessing.Radon3D(t, hy, hx, py, px, centeredh=True,
                                       kind='parabolic', interp=False,
                                       engine='numpy')
RHop = pylops.signalprocessing.Radon3D(t, hy, hx, py, px, centeredh=True,
                                       kind='hyperbolic', interp=False,
                                       engine='numpy')

# forward
yL = RLop * x.ravel()
yP = RPop * x.ravel()
yH = RHop * x.ravel()
yL = yL.reshape(ny, nx, nt)
yP = yP.reshape(ny, nx, nt)
yH = yH.reshape(ny, nx, nt)

# adjoint
xadjL = RLop.H * yL.ravel()
xadjP = RPop.H * yP.ravel()
xadjH = RHop.H * yH.ravel()
xadjL = xadjL.reshape(npy, npx, nt)
xadjP = xadjP.reshape(npy, npx, nt)
xadjH = xadjH.reshape(npy, npx, nt)

# plotting
fig, axs = plt.subplots(2, 4, figsize=(10, 6))
axs[0][0].imshow(x[npy//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[0][0].set_title('Input model')
axs[0][0].axis('tight')
axs[0][1].imshow(yL[ny//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hx[0], hx[-1], t[-1], t[0]))
axs[0][1].set_title('Linear data')
axs[0][1].axis('tight')
axs[0][2].imshow(yP[ny//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hx[0], hx[-1], t[-1], t[0]))
axs[0][2].set_title('Parabolic data')
axs[0][2].axis('tight')
axs[0][3].imshow(yH[ny//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hx[0], hx[-1], t[-1], t[0]))
axs[0][3].set_title('Hyperbolic data')
axs[0][3].axis('tight')
axs[1][1].imshow(xadjL[npy//2].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][0].axis('off')
axs[1][1].set_title('Linear adjoint')
axs[1][1].axis('tight')
axs[1][2].imshow(xadjP[npy//2].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][2].set_title('Parabolic adjoint')
axs[1][2].axis('tight')
axs[1][3].imshow(xadjH[npy//2].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(px[0], px[-1], t[-1], t[0]))
axs[1][3].set_title('Hyperbolic adjoint')
axs[1][3].axis('tight')
fig.tight_layout()

fig, axs = plt.subplots(2, 4, figsize=(10, 6))
axs[0][0].imshow(x[:, npx//2-2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(py[0], py[-1], t[-1], t[0]))
axs[0][0].set_title('Input model')
axs[0][0].axis('tight')
axs[0][1].imshow(yL[:, nx//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hy[0], hy[-1], t[-1], t[0]))
axs[0][1].set_title('Linear data')
axs[0][1].axis('tight')
axs[0][2].imshow(yP[:, nx//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hy[0], hy[-1], t[-1], t[0]))
axs[0][2].set_title('Parabolic data')
axs[0][2].axis('tight')
axs[0][3].imshow(yH[:, nx//2].T, vmin=-1, vmax=1, cmap='seismic_r',
                 extent=(hy[0], hy[-1], t[-1], t[0]))
axs[0][3].set_title('Hyperbolic data')
axs[0][3].axis('tight')
axs[1][1].imshow(xadjL[:, npx//2-5].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(py[0], py[-1], t[-1], t[0]))
axs[1][0].axis('off')
axs[1][1].set_title('Linear adjoint')
axs[1][1].axis('tight')
axs[1][2].imshow(xadjP[:, npx//2-2].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(py[0], py[-1], t[-1], t[0]))
axs[1][2].set_title('Parabolic adjoint')
axs[1][2].axis('tight')
axs[1][3].imshow(xadjH[:, npx//2-2].T, vmin=-100, vmax=100, cmap='seismic_r',
                 extent=(py[0], py[-1], t[-1], t[0]))
axs[1][3].set_title('Hyperbolic adjoint')
axs[1][3].axis('tight')
fig.tight_layout()
Radon Transform =============== This example shows how to use the :py:class:`pylops.signalprocessing.Radon2D` and :py:class:`pylops.signalprocessing.Radon3D` operators to apply the Radon Transform to 2-dimensional or 3-dimensional signals, respectively. In our implementation both linear, parabolic and hyperbolic parametrization can be chosen. ############################################################################### # Let's start by creating an empty 2d matrix of size :math:`n_{p_x} \times n_t` # and add a single spike in it. We will see that applying the forward # Radon operator will result in a single event (linear, parabolic or # hyperbolic) in the resulting data vector. ############################################################################### # We can now define our operators for different parametric curves and apply # them to the input model vector. We also apply the adjoint to the resulting # data vector. # forward # adjoint ############################################################################### # Let's now visualize the input model in the Radon domain, the data, and # the adjoint model the different parametric curves. ############################################################################### # As we can see in the bottom figures, the adjoint Radon transform is far # from being close to the inverse Radon transform, i.e. # :math:`\mathbf{R^H}\mathbf{R} \neq \mathbf{I}` (compared to the case of FFT # where the adjoint and inverse are equivalent, i.e. # :math:`\mathbf{F^H}\mathbf{F} = \mathbf{I}`). In fact when we apply the # adjoint Radon Transform we obtain a *model* that # is a smoothed version of the original model polluted by smearing and # artifacts. In tutorial :ref:`sphx_glr_tutorials_radonfiltering.py` we will # exploit a sparsity-promiting Radon transform to perform filtering of unwanted # signals from an input data. # # Finally we repeat the same exercise with 3d data. # forward # adjoint # plotting