import numpy as np
from pylops.utils.typing import NDArray
[docs]
class EuclideanBallProj:
r"""Euclidean ball projection.
Parameters
----------
center : :obj:`numpy.ndarray` or :obj:`float`
Center of the ball
radius : :obj:`float`
Radius
Notes
-----
Given an Euclidean ball defined as:
.. math::
\operatorname{Eucl}_{[c, r]} = \{ \mathbf{x}: l ||\mathbf{x} - \mathbf{c}||_2 \leq r \}
its orthogonal projection is:
.. math::
P_{\operatorname{Eucl}_{[c, r]}} (\mathbf{x}) = \mathbf{c} + \frac{r}
{max\{ ||\mathbf{x} - \mathbf{c}||_2^2, r\}}(\mathbf{x} - \mathbf{c})
Note the this is the proximal operator of the corresponding
indicator function :math:`\mathcal{I}_{\operatorname{Eucl}_{[c, r]}}`.
"""
def __init__(self, center: NDArray | float, radius: float):
self.center = center
self.radius = radius
def __call__(self, x: NDArray) -> NDArray:
x = self.center + self.radius / (
max(float(np.linalg.norm(x - self.center)), self.radius)
) * (x - self.center)
return x
PyProximal