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Low Resolution Electromagnetic Tomography

Solver ID: LORETA

Usage

from invert import Solver

# fwd = ...    (mne.Forward object)
# evoked = ... (mne.Evoked object)

solver = Solver("LORETA")
solver.make_inverse_operator(fwd)
stc = solver.apply_inverse_operator(evoked)
stc.plot()

Overview

Smoothness-constrained minimum-norm inverse that penalizes spatial roughness via a Laplacian operator on the source space.

References

  1. Pascual-Marqui, R. D., Michel, C. M., & Lehmann, D. (1994). Low resolution electromagnetic tomography: a new method for localizing electrical activity in the brain. International Journal of Psychophysiology, 18(1), 49–65.

API Reference

Bases: BaseSolver

Class for the Low Resolution Tomography (LORETA) inverse solution.

References

[1] Pascual-Marqui, R. D. (1999). Review of methods for solving the EEG inverse problem. International journal of bioelectromagnetism, 1(1), 75-86.

Source code in invert/solvers/minimum_norm/loreta.py 
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class SolverLORETA(BaseSolver):
    """Class for the Low Resolution Tomography (LORETA) inverse solution.

    References
    ----------
    [1] Pascual-Marqui, R. D. (1999). Review of methods for solving the EEG
    inverse problem. International journal of bioelectromagnetism, 1(1), 75-86.

    """

    meta = SolverMeta(
        acronym="LORETA",
        full_name="Low Resolution Electromagnetic Tomography",
        category="Minimum Norm",
        description=(
            "Smoothness-constrained minimum-norm inverse that penalizes spatial "
            "roughness via a Laplacian operator on the source space."
        ),
        references=[
            "Pascual-Marqui, R. D., Michel, C. M., & Lehmann, D. (1994). Low resolution electromagnetic tomography: a new method for localizing electrical activity in the brain. International Journal of Psychophysiology, 18(1), 49–65.",
        ],
    )

    def __init__(self, name="Low Resolution Tomography", **kwargs):
        self.name = name
        return super().__init__(**kwargs)

    def make_inverse_operator(self, forward, *args, alpha="auto", **kwargs):
        """Calculate inverse operator.

        Parameters
        ----------
        forward : mne.Forward
            The mne-python Forward model instance.
        alpha : float
            The regularization parameter.

        Return
        ------
        self : object returns itself for convenience
        """
        super().make_inverse_operator(forward, *args, alpha=alpha, **kwargs)
        leadfield = self.leadfield
        LTL = leadfield.T @ leadfield
        B = np.eye(leadfield.shape[1])
        adjacency = mne.spatial_src_adjacency(
            forward["src"], verbose=self.verbose
        ).toarray()
        laplace_operator = laplacian(adjacency)
        BLapTLapB = B @ laplace_operator.T @ laplace_operator @ B

        if alpha == "auto":
            r_grid = np.asarray(self.r_values, dtype=float)
        else:
            r_grid = np.asarray([float(alpha)], dtype=float)
        max_eig_LTL = float(np.linalg.svd(LTL, compute_uv=False).max())
        max_eig_penalty = float(np.linalg.svd(BLapTLapB, compute_uv=False).max())
        scale = max_eig_LTL / max(max_eig_penalty, 1e-15)
        self.alphas = list(r_grid * scale)

        inverse_operators = []
        for alpha in self.alphas:
            inverse_operator = np.linalg.inv(LTL + (alpha) * BLapTLapB) @ leadfield.T
            inverse_operators.append(inverse_operator)

        self.inverse_operators = [
            InverseOperator(inverse_operator, self.name)
            for inverse_operator in inverse_operators
        ]
        return self

__init__ 

__init__(name='Low Resolution Tomography', **kwargs)
Source code in invert/solvers/minimum_norm/loreta.py 
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def __init__(self, name="Low Resolution Tomography", **kwargs):
    self.name = name
    return super().__init__(**kwargs)

make_inverse_operator 

make_inverse_operator(
    forward, *args, alpha="auto", **kwargs
)

Calculate inverse operator.

Parameters:

Name Type Description Default
forward Forward

The mne-python Forward model instance.

 required
alpha float

The regularization parameter.

 'auto'
Return

self : object returns itself for convenience

Source code in invert/solvers/minimum_norm/loreta.py 
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def make_inverse_operator(self, forward, *args, alpha="auto", **kwargs):
    """Calculate inverse operator.

    Parameters
    ----------
    forward : mne.Forward
        The mne-python Forward model instance.
    alpha : float
        The regularization parameter.

    Return
    ------
    self : object returns itself for convenience
    """
    super().make_inverse_operator(forward, *args, alpha=alpha, **kwargs)
    leadfield = self.leadfield
    LTL = leadfield.T @ leadfield
    B = np.eye(leadfield.shape[1])
    adjacency = mne.spatial_src_adjacency(
        forward["src"], verbose=self.verbose
    ).toarray()
    laplace_operator = laplacian(adjacency)
    BLapTLapB = B @ laplace_operator.T @ laplace_operator @ B

    if alpha == "auto":
        r_grid = np.asarray(self.r_values, dtype=float)
    else:
        r_grid = np.asarray([float(alpha)], dtype=float)
    max_eig_LTL = float(np.linalg.svd(LTL, compute_uv=False).max())
    max_eig_penalty = float(np.linalg.svd(BLapTLapB, compute_uv=False).max())
    scale = max_eig_LTL / max(max_eig_penalty, 1e-15)
    self.alphas = list(r_grid * scale)

    inverse_operators = []
    for alpha in self.alphas:
        inverse_operator = np.linalg.inv(LTL + (alpha) * BLapTLapB) @ leadfield.T
        inverse_operators.append(inverse_operator)

    self.inverse_operators = [
        InverseOperator(inverse_operator, self.name)
        for inverse_operator in inverse_operators
    ]
    return self