L1Norm¶

class odl.solvers.functional.default_functionals.L1Norm(*args, **kwargs)[source]¶

The functional corresponding to L1-norm.

The L1-norm, ||x||_1, is defined as the integral/sum of |x|.

Notes

If the functional is defined on an $\mathbb{R}^n$ -like space, the $\| \cdot \|_1$ -norm is defined as

$\| x \|_1 = \sum_{i=1}^n |x_i|.$

If the functional is defined on an $L_2$ -like space, the $\| \cdot \|_1$ -norm is defined as

$\| x \|_1 = \int_\Omega |x(t)| dt.$

The proximal factory allows using vector-valued stepsizes:

>>> space = odl.rn(3)
>>> f = odl.solvers.L1Norm(space)
>>> x = space.one()
>>> f.proximal([0.5, 1.0, 1.5])(x)
rn(3).element([ 0.5,  0. ,  0. ])

Attributes

Methods

`_call`(self, x)	Return the Lp-norm of `x`.
`bregman`(self, point, subgrad)	Return the Bregman distance functional.
`derivative`(self, point)	Return the derivative operator in the given point.
`norm`(self[, estimate])	Return the operator norm of this operator.
`translated`(self, shift)	Return a translation of the functional.

__init__(self, space)[source]¶

Initialize a new instance.

Parameters