pyproximal.RelaxedMumfordShah

class pyproximal.RelaxedMumfordShah(sigma: Union[float, ndarray[tuple[Any, ...], dtype[_ScalarT]], Callable[[int], Union[float, ndarray[tuple[Any, ...], dtype[_ScalarT]]]]] = 1.0, kappa: Union[float, ndarray[tuple[Any, ...], dtype[_ScalarT]], Callable[[int], Union[float, ndarray[tuple[Any, ...], dtype[_ScalarT]]]]] = 1.0)

Relaxed Mumford-Shah norm proximal operator.

Proximal operator of the relaxed Mumford-Shah norm: \(\text{rMS}(x) = \min (\alpha\Vert x\Vert_2^2, \kappa)\).

Parameters

sigmafloat or np.ndarray or func, optional

Multiplicative coefficient of L2 norm that controls the smoothness of the solutuon. This can be a constant number, a list of values (for multidimensional inputs, acting on the second dimension) 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 a scalar (or a list of) sigma to be used.

kappafloat or np.ndarray or func, optional

Constant value in the rMS norm which essentially controls when the norm allows a jump. This can be a constant number, a list of values (for multidimensional inputs, acting on the second dimension) 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 a scalar (or a list of) kappa to be used.

Notes

The \(rMS\) proximal operator is defined as [1]:

\[\begin{split}\text{prox}_{\tau \text{rMS}}(x) = \begin{cases} \frac{1}{1+2\tau\alpha}x & \text{ if } & \vert x\vert \leq \sqrt{\frac{\kappa}{\alpha}(1 + 2\tau\alpha)} \\ \kappa & \text{ else } \end{cases}.\end{split}\]

1

Strekalovskiy, E., and D. Cremers, 2014, Real-time minimization of the piecewise smooth Mumford-Shah functional: European Conference on Computer Vision, 127–141.

Methods

__init__([sigma, kappa])
affine_addition(v)	Affine addition
chain(g)	Chain
grad(x)	Gradient of the Moreau envelope of the function.
postcomposition(sigma)	Postcomposition
precomposition(a, b)	Precomposition
prox(*args, **kwargs)
proxdual(**kwargs)

Examples using pyproximal.RelaxedMumfordShah

Relaxed Mumford-Shah regularization

Relaxed Mumford-Shah regularization