mlem

odl.solvers.iterative.statistical.mlem(op, x, data, niter, callback=None, **kwargs)[source]

Maximum Likelihood Expectation Maximation algorithm.

Attempts to solve:

max_x L(x | data)

where L(x | data) is the Poisson likelihood of x given data. The likelihood depends on the forward operator op such that (approximately):

op(x) = data

Parameters

opOperator

Forward operator in the inverse problem.

xop.domain element

Vector to which the result is written. Its initial value is used as starting point of the iteration, and its values are updated in each iteration step. The initial value of x should be non-negative.

dataop.range element-like

Right-hand side of the equation defining the inverse problem.

niterint

Number of iterations.

callbackcallable, optional

Function called with the current iterate after each iteration.

Other Parameters

sensitivitiesfloat or op.domain element-like, optional

The algorithm contains a A^T 1 term, if this parameter is given, it is replaced by it. Default: op.adjoint(op.range.one())

Notes

Given a forward model A and data g, the algorithm attempts to find an x that maximizes:

P(g | g \text{ is } Poisson(A(x)) \text{ distributed}).

The algorithm is explicitly given by:

x_{n+1} = \frac{x_n}{A^* 1} A^* (g / A(x_n))

See Also

osmlem : Ordered subsets MLEM loglikelihood : Function for calculating the logarithm of the likelihood