axs[1].axis('tight')
axs[1].set_title('y sym')
plt.colorbar(im, ax=axs[1])
im = axs[2].imshow(Basym, interpolation='nearest', cmap='rainbow')
axs[2].axis('tight')
axs[2].set_title('y asym')
plt.colorbar(im, ax=axs[2])
plt.tight_layout()
plt.subplots_adjust(top=0.8)
###############################################################################
# We consider now the gradient operator. Given a 2-dimensional array,
# this operator applies first-order derivatives on both dimensions and
# concatenates them.
Gop = pylops.Gradient(dims=(nx, ny), dtype='float64')
B = np.reshape(Gop * A.flatten(), (2*nx, ny))
C = np.reshape(Gop.H * B.flatten(), (nx, ny))
fig, axs = plt.subplots(1, 3, figsize=(10, 3))
fig.suptitle('Gradient', fontsize=12,
fontweight='bold', y=0.95)
im = axs[0].imshow(A, interpolation='nearest', cmap='rainbow')
axs[0].axis('tight')
axs[0].set_title('x')
plt.colorbar(im, ax=axs[0])
im = axs[1].imshow(B, interpolation='nearest', cmap='rainbow')
axs[1].axis('tight')
axs[1].set_title('y')
plt.colorbar(im, ax=axs[1])
# Load image
img = camera()
ny, nx = img.shape
# Add noise
sigman = 20
n = np.random.normal(0, sigman, img.shape)
noise_img = img + n
###############################################################################
# We can now define a :class:`pylops.Gradient` operator as well as the
# different proximal operators to be passed to our solvers
# Gradient operator
sampling = 1.
Gop = pylops.Gradient(dims=(ny, nx), sampling=sampling, edge=False,
kind='forward', dtype='float64')
L = 8. / sampling ** 2 # maxeig(Gop^H Gop)
# L2 data term
lamda = .04
l2 = pyproximal.L2(b=noise_img.ravel(), sigma=lamda)
# L1 regularization (isotropic TV)
l1iso = pyproximal.L21(ndim=2)
###############################################################################
# To start, we solve our denoising problem with the original Primal-Dual
# algorithm
# Primal-dual
tau = 0.95 / np.sqrt(L)
def Gradient(shape: tuple, step: Union[tuple, float] = 1., edge: bool = True, dtype: str = 'float64',
kind: str = 'centered') -> PyLopLinearOperator:
r"""
Gradient.
Computes the gradient of a multi-dimensional array (at least two dimensions are required).
Parameters
----------
shape: tuple
Shape of the input array.
step: Union[float, Tuple[float, ...]]
Step size in each direction.
edge: bool
For ``kind = 'centered'``, use reduced order derivative at edges (``True``) or ignore them (``False``).
dtype: str
Type of elements in input vector.
kind: str
Derivative kind (``forward``, ``centered``, or ``backward``).
Returns
-------
:py:class:`pycsou.core.linop.LinearOperator`
Gradient operator.
Examples
--------
.. testsetup::
import numpy as np
from pycsou.linop.diff import Gradient, FirstDerivative
from pycsou.util.misc import peaks
.. doctest::
>>> x = np.linspace(-2.5, 2.5, 100)
>>> X,Y = np.meshgrid(x,x)
>>> Z = peaks(X, Y)
>>> Nabla = Gradient(shape=Z.shape, kind='forward')
>>> D = FirstDerivative(size=Z.size, shape=Z.shape, kind='forward')
>>> np.allclose((Nabla * Z.flatten())[:Z.size], D * Z.flatten())
True
.. plot::
import numpy as np
import matplotlib.pyplot as plt
from pycsou.linop.diff import Gradient
from pycsou.util.misc import peaks
x = np.linspace(-2.5, 2.5, 25)
X,Y = np.meshgrid(x,x)
Z = peaks(X, Y)
Dop = Gradient(shape=Z.shape)
y = Dop * Z.flatten()
plt.figure()
h = plt.pcolormesh(X,Y,Z, shading='auto')
plt.colorbar(h)
plt.title('Signal')
plt.figure()
h = plt.pcolormesh(X,Y,y[:Z.size].reshape(X.shape), shading='auto')
plt.colorbar(h)
plt.title('Gradient (1st component)')
plt.figure()
h = plt.pcolormesh(X,Y,y[Z.size:].reshape(X.shape), shading='auto')
plt.colorbar(h)
plt.title('Gradient (2nd component)')
plt.show()
Notes
-----
The ``Gradient`` operator applies a first-order derivative to each dimension of
a multi-dimensional array in forward mode.
For simplicity, given a three dimensional array, the ``Gradient`` in forward
mode using a centered stencil can be expressed as:
.. math::
\mathbf{g}_{i, j, k} =
(f_{i+1, j, k} - f_{i-1, j, k}) / d_1 \mathbf{i_1} +
(f_{i, j+1, k} - f_{i, j-1, k}) / d_2 \mathbf{i_2} +
(f_{i, j, k+1} - f_{i, j, k-1}) / d_3 \mathbf{i_3}
which is discretized as follows:
.. math::
\mathbf{g} =
\begin{bmatrix}
\mathbf{df_1} \\
\mathbf{df_2} \\
\mathbf{df_3}
\end{bmatrix}.
In adjoint mode, the adjoints of the first derivatives along different
axes are instead summed together.
See Also
--------
:py:func:`~pycsou.linop.diff.DirectionalGradient`, :py:func:`~pycsou.linop.diff.FirstDerivative`
"""
return PyLopLinearOperator(pylops.Gradient(dims=shape, sampling=step, edge=edge, dtype=dtype, kind=kind))
def multi_class_segmentation(img,
classes: list,
beta: float = 0.001,
tau: float = None):
raveld_f = np.zeros(((np.prod(img.shape), len(classes))))
for i in range(len(classes)):
raveld_f[:, i] = (img.ravel() - classes[i])**2
f = raveld_f
grad = pylops.Gradient(dims=img.shape,
edge=True,
dtype='float64',
kind="backward")
K = beta * grad
G = DatatermLinear()
G.set_proxdata(f)
F_star = Projection(f.shape, len(img.shape))
solver = PdHgm(K, F_star, G)
solver.var['x'] = np.zeros((K.shape[1], len(classes)))
solver.var['y'] = np.zeros((K.shape[0], len(classes)))
if tau:
tau0 = tau
else:
norm = np.abs(np.asscalar(K.eigs(neigs=1, which='LM')))
tau0 = 0.99 / norm
print(
"Calced tau: " + str(tau0) + ". "
"Next run with same beta, set this tau value to decrease runtime.")
sigma0 = tau0
G.set_proxparam(tau0)
F_star.set_proxparam(sigma0)
solver.maxiter = 200
solver.tol = 10**(-6)
solver.solve()
seg = np.reshape(solver.var['x'],
tuple(list(img.shape) + [len(classes)]),
order='C')
a = seg
result = [] #np.zeros(a.shape)
#for i in range(a.shape[0]): # SLOW
# for j in range(a.shape[1]):
# idx = np.argmin((a[i, j, :]))
# result[i, j, idx] = 1
tmp_result = np.argmin(a, axis=len(img.shape))
for i, c in enumerate(classes):
result.append((tmp_result == i).astype(int))
result0 = sum([i * result[i] for i in range(len(classes))])
result = np.array(result)
return result0, result
def solve(self, f: np.ndarray):
"""
Description of main primal-dual iteration.
:return: None
"""
(primal_n, primal_m) = self.image_size
v = w = 0
g = f.ravel()
p = p_bar = np.zeros(primal_n*primal_m)
q = q_bar = np.zeros(2*primal_n*primal_m)
if self.reg_mode != 'tik':
grad = pylops.Gradient(dims=(primal_n, primal_m), dtype='float64', edge=True, kind="backward")
else:
grad = pylops.Identity(np.prod(self.image_size))
grad1 = pylops.BlockDiag([grad, grad]) # symmetric dxdy <-> dydx not necessary (expensive) but easy and functional
proj_0 = IndicatorL2((primal_n, primal_m), upper_bound=self.alpha[0])
proj_1 = IndicatorL2((2 * primal_n, primal_m), upper_bound=self.alpha[1])
if not self.silent_mode:
progress = progressbar.ProgressBar(max_value=self.max_iter)
k = 0
while (self.tol < self.sens or k == 0) and (k < self.max_iter):
p_old = p
q_old = q
# Dual Update
g = self.lam / (self.tau + self.lam) * (g + self.tau * (self.A*(p_bar ) - f)) #- self.alpha[0]*self.breg_p
if self.reg_mode != 'tik':
v = proj_0.prox(v + self.tau * (grad * p_bar - q_bar))
else:
v = self.alpha[0] / (self.tau + self.alpha[0]) * \
(v + self.tau * (grad*p_bar - self.data))
if self.reg_mode == 'tgv':
w = proj_1.prox(w + self.tau * grad1*q_bar)
# Primal Update
p = p - self.tau * (-self.alpha[0]*self.breg_p + self.A.H*g + grad.H * v)
if self.reg_mode == 'tgv':
q = q + self.tau * (v - grad1.H * w)
# Extragradient Update
p_bar = 2 * p - p_old
q_bar = 2 * q - q_old
if k % 50 == 0:
p_gap = p - p_old
self.sens = np.linalg.norm(p_gap)/np.linalg.norm(p_old)
print(self.sens)
if self.gamma:
raise NotImplementedError("The adjustment of the step size in the "
"Primal-Dual is not yet fully developed.")
thetha = 1 / np.sqrt(1 + 2*self.gamma * self.G.prox_param)
self.G.prox_param = thetha * self.G.prox_param
self.F_star.prox_param = self.F_star.prox_param / thetha
k += 1
if not self.silent_mode:
progress.update(k)
self.x = p
if k <= self.max_iter:
print(" Early stopping.")
return self.x
def multi_class_segmentation(img,
classes: list,
beta: float = 0.001,
tau: float = None):
#f = np.zeros( tuple( list(img.shape) + [len(classes)]))
raveld_f = np.zeros(((np.prod(img.shape), len(classes))))
for i in range(len(classes)):
#f[:, :, i] = (img.T - classes[i]) ** 2
raveld_f[:, i] = (img.ravel() - classes[i])**2
#f = np.ravel(f, order='C')
f = raveld_f
grad = pylops.Gradient(dims=img.shape, dtype='float64')
# grad = FirstDerivative(262144, boundaries=boundaries)
K = beta * grad
# vd1 = convex_segmentation(u0, beta0, classes)
G = DatatermLinear()
G.set_proxdata(f)
F_star = Projection(f.shape, len(img.shape))
solver = PdHgm(K, F_star, G)
solver.var['x'] = np.zeros((K.shape[1], len(classes)))
solver.var['y'] = np.zeros((K.shape[0], len(classes)))
if tau:
tau0 = tau
else:
tau0 = 0.99 / normest(K)
print(tau0)
sigma0 = tau0
G.set_proxparam(tau0)
F_star.set_proxparam(sigma0)
solver.maxiter = 200
solver.tol = 10**(-6)
# G.set_proxdata(f)
solver.solve()
seg = np.reshape(solver.var['x'],
tuple(list(img.shape) + [len(classes)]),
order='C')
a = seg
result = [] #np.zeros(a.shape)
#for i in range(a.shape[0]): # SLOW
# for j in range(a.shape[1]):
# idx = np.argmin((a[i, j, :]))
# result[i, j, idx] = 1
tmp_result = np.argmin(a, axis=len(img.shape))
for i, c in enumerate(classes):
result.append((tmp_result == i).astype(int))
result0 = sum([i * result[i] for i in range(len(classes))])
result = np.array(result)
return result0, result