from collections.abc import Callable
from copy import deepcopy
from typing import Any
import numpy as np
from pylops import FirstDerivative, Gradient
from pylops.utils.typing import NDArray, ShapeLike
from typing_extensions import Self
from pyproximal.ProxOperator import ProxOperator, _check_tau
[docs]
class TV(ProxOperator):
r"""TV Norm proximal operator.
Proximal operator for the TV norm defined as: :math:`f(\mathbf{x}) =
\sigma ||\mathbf{x}||_{\text{TV}}`.
Parameters
----------
dims : :obj:`tuple`, optional
Number of samples for each dimension
(``None`` if only one dimension is available)
sigma : :obj:`float`, optional
Multiplicative coefficient of TV norm
niter : :obj:`int` or :obj:`func`, optional
Number of iterations of iterative scheme used to compute the proximal.
This can be a constant number or a function that is called passing a
counter which keeps track of how many times the ``prox`` method has
been invoked before and returns the ``niter`` to be used.
rtol : :obj:`float`, optional
Relative tolerance for stopping criterion.
Notes
-----
The proximal algorithm is implemented following [1]_.
.. [1] Beck, A. and Teboulle, M., "Fast gradient-based algorithms for constrained total
variation image denoising and deblurring problems", 2009.
"""
def __init__(
self,
dims: ShapeLike | None,
sigma: float = 1.0,
niter: int | Callable[[int], int] = 10,
rtol: float = 1e-4,
**kwargs: Any,
) -> None:
super().__init__(None, True)
self.dims = (0,) if dims is None else dims
self.ndim = 1 if dims is None else len(dims)
self.sigma = sigma
self.niter = niter
self.count = 0
self.rtol = rtol
self.kwargs = kwargs
def __call__(self, x: NDArray) -> float:
x = x.reshape(self.dims)
if self.ndim == 1:
derivOp = FirstDerivative(
dims=self.dims[0], axis=0, edge=False, dtype=x.dtype, kind="forward"
)
dx = derivOp @ x
y = np.sum(np.abs(dx), axis=0)
elif self.ndim >= 2:
y = 0
gradOp = Gradient(self.dims, edge=False, dtype=x.dtype, kind="forward")
grads = gradOp.matvec(x.ravel())
grads = grads.reshape((self.ndim,) + self.dims)
for g in grads:
y += np.power(abs(g), 2)
y = np.sqrt(y)
return float(self.sigma * np.sum(y))
def _increment_count(func: Callable[..., Any]) -> Callable[..., Any]:
"""Increment counter"""
def wrapped(self: Self, *args: Any, **kwargs: Any) -> Any:
self.count += 1
return func(self, *args, **kwargs)
return wrapped
@_increment_count
@_check_tau
def prox(self, x: NDArray, tau: float) -> NDArray:
# define current number of iterations
if isinstance(self.niter, int):
niter = self.niter
else:
niter = self.niter(self.count)
gamma = self.sigma * tau
rtol = self.rtol
# TODO implement test_gamma
# Initialization
x = x.reshape(self.dims)
sol = x
if self.ndim == 1:
derivOp = FirstDerivative(
dims=self.dims[0], axis=0, edge=False, dtype=x.dtype, kind="forward"
)
else:
gradOp = Gradient(x.shape, edge=False, dtype=x.dtype, kind="forward")
if self.ndim == 1:
r = derivOp @ (x * 0)
rr = deepcopy(r)
elif self.ndim == 2:
r, s = gradOp.matvec((x * 0).ravel()).reshape((self.ndim,) + x.shape)
rr, ss = deepcopy(r), deepcopy(s)
elif self.ndim == 3:
r, s, k = gradOp.matvec((x * 0).ravel()).reshape((self.ndim,) + x.shape)
rr, ss, kk = deepcopy(r), deepcopy(s), deepcopy(k)
elif self.ndim == 4:
r, s, k, u = gradOp.matvec((x * 0).ravel()).reshape((self.ndim,) + x.shape)
rr, ss, kk, uu = deepcopy(r), deepcopy(s), deepcopy(k), deepcopy(u)
if self.ndim >= 1:
pold = r
if self.ndim >= 2:
qold = s
if self.ndim >= 3:
oold = k
if self.ndim >= 4:
mold = u
told, prev_obj = 1.0, 0.0
# Initialization for weights
if self.ndim >= 1:
try:
wx = self.kwargs["wx"]
except (KeyError, TypeError):
wx = 1.0
if self.ndim >= 2:
try:
wy = self.kwargs["wy"]
except (KeyError, TypeError):
wy = 1.0
if self.ndim >= 3:
try:
wz = self.kwargs["wz"]
except (KeyError, TypeError):
wz = 1.0
if self.ndim >= 4:
try:
wt = self.kwargs["wt"]
except (KeyError, TypeError):
wt = 1.0
if self.ndim == 1:
mt = wx
elif self.ndim == 2:
mt = np.maximum(wx, wy)
elif self.ndim == 3:
mt = np.maximum(wx, np.maximum(wy, wz))
elif self.ndim == 4:
mt = np.maximum(np.maximum(wx, wy), np.maximum(wz, wt))
if self.ndim >= 1:
try:
rr *= np.conjugate(wx)
except KeyError:
pass
if self.ndim >= 2:
try:
ss *= np.conjugate(wy)
except KeyError:
pass
if self.ndim >= 3:
try:
kk *= np.conjugate(wz)
except KeyError:
pass
if self.ndim >= 4:
try:
uu *= np.conjugate(wt)
except KeyError:
pass
iter = 0
while iter <= niter:
# Current Solution
if self.ndim >= 1:
div = np.concatenate(
(
np.expand_dims(
rr[0,],
axis=0,
),
rr[1:-1,] - rr[:-2,],
-np.expand_dims(
rr[-2,],
axis=0,
),
),
axis=0,
)
if self.ndim >= 2:
div += np.concatenate(
(
np.expand_dims(
ss[
:,
0,
],
axis=1,
),
ss[
:,
1:-1,
]
- ss[
:,
:-2,
],
-np.expand_dims(
ss[
:,
-2,
],
axis=1,
),
),
axis=1,
)
if self.ndim >= 3:
div += np.concatenate(
(
np.expand_dims(
kk[
:,
:,
0,
],
axis=2,
),
kk[
:,
:,
1:-1,
]
- kk[
:,
:,
:-2,
],
-np.expand_dims(
kk[
:,
:,
-2,
],
axis=2,
),
),
axis=2,
)
if self.ndim >= 4:
div += np.concatenate(
(
np.expand_dims(
uu[
:,
:,
:,
0,
],
axis=3,
),
uu[
:,
:,
:,
1:-1,
]
- uu[
:,
:,
:,
:-2,
],
-np.expand_dims(
uu[
:,
:,
:,
-2,
],
axis=3,
),
),
axis=3,
)
sol = x - gamma * div
# Objective function value
obj = 0.5 * np.power(np.linalg.norm(x[:] - sol[:]), 2) + gamma * np.sum(
self.__call__(sol), axis=0
)
if obj > 1e-10:
rel_obj = np.abs(obj - prev_obj) / obj
else:
rel_obj = 2 * rtol
prev_obj = obj
# Stopping criterion
if rel_obj < rtol:
break
# Update divergence vectors and project
if self.ndim == 1:
dx = derivOp @ sol
r -= 1.0 / (4 * gamma * mt**2) * dx
weights = np.maximum(1, np.abs(r))
elif self.ndim == 2:
dx, dy = gradOp.matvec(sol.ravel()).reshape((self.ndim,) + x.shape)
r -= (1.0 / (8.0 * gamma * mt**2.0)) * dx
s -= (1.0 / (8.0 * gamma * mt**2.0)) * dy
weights = np.maximum(
1, np.sqrt(np.power(np.abs(r), 2) + np.power(np.abs(s), 2))
)
elif self.ndim == 3:
dx, dy, dz = gradOp.matvec(sol.ravel()).reshape((self.ndim,) + x.shape)
r -= 1.0 / (12.0 * gamma * mt**2) * dx
s -= 1.0 / (12.0 * gamma * mt**2) * dy
k -= 1.0 / (12.0 * gamma * mt**2) * dz
weights = np.maximum(
1,
np.sqrt(
np.power(np.abs(r), 2)
+ np.power(np.abs(s), 2)
+ np.power(np.abs(k), 2)
),
)
elif self.ndim == 4:
dx, dy, dz, dt = gradOp.matvec(sol.ravel()).reshape(
(self.ndim,) + x.shape
)
r -= 1.0 / (16 * gamma * mt**2) * dx
s -= 1.0 / (16 * gamma * mt**2) * dy
k -= 1.0 / (16 * gamma * mt**2) * dz
u -= 1.0 / (16 * gamma * mt**2) * dt
weights = np.maximum(
1,
np.sqrt(
np.power(np.abs(r), 2)
+ np.power(np.abs(s), 2)
+ np.power(np.abs(k), 2)
+ np.power(np.abs(u), 2)
),
)
# FISTA update
t = (1 + np.sqrt(4 * told**2)) / 2.0
if self.ndim >= 1:
p = r / weights
r = p + (told - 1) / t * (p - pold)
pold = p
rr = deepcopy(r)
if self.ndim >= 2:
q = s / weights
s = q + (told - 1) / t * (q - qold)
qold = q
ss = deepcopy(s)
if self.ndim >= 3:
o = k / weights
k = o + (told - 1) / t * (o - oold)
oold = o
kk = deepcopy(k)
if self.ndim >= 4:
m = u / weights
u = m + (told - 1) / t * (m - mold)
mold = m
uu = deepcopy(u)
told = t
iter += 1
return sol.ravel()
PyProximal