Initial commit
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redclawsystems
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//! eLORETA (Exact Low Resolution Electromagnetic Tomography) inverse solution.
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//!
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//! eLORETA is an inverse method with the unique property of exact (zero-error)
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//! localization for point sources. Unlike sLORETA which only standardizes the
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//! current density, eLORETA computes weights that ensure the resolution matrix
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//! has specific properties guaranteeing zero localization error.
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//!
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//! ## Mathematical Background
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//!
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//! The eLORETA solution is: J = W * M
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//!
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//! where W is computed iteratively such that the resulting resolution matrix
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//! R = W * G satisfies specific properties.
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//!
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//! The weight matrix W_i for source i is:
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//! W_i = [G_i^T * C^(-1) * G_i]^(-1/2) for each source orientation block
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//!
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//! This ensures that the point-spread function is centered on the true source.
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//!
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//! ## References
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//!
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//! - Pascual-Marqui, R.D. (2007). Discrete, 3D distributed, linear imaging methods
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//! of electric neuronal activity. Part 1: exact, zero error localization.
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//! - Pascual-Marqui, R.D. (2009). Theory of the EEG inverse problem.
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//!
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//! ## Usage
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//!
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//! ```rust,ignore
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//! use rtx_neuro_inverse::loreta::{EloretaInverse, EloretaConfig};
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//!
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//! let config = EloretaConfig::default();
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//! let inverse = EloretaInverse::make_inverse(&gain, &noise_cov, config)?;
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//! let stc = inverse.apply(&evoked_data)?;
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//! ```
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use crate::{Covariance, InverseError, InverseResult, SourceEstimate};
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use nalgebra::{DMatrix, DVector, Matrix3};
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use rtx_neuro_forward::GainMatrix;
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/// Configuration for eLORETA inverse
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#[derive(Debug, Clone)]
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pub struct EloretaConfig {
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/// Regularization parameter (lambda^2)
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pub lambda2: f64,
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/// Maximum iterations for weight computation
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pub max_iter: usize,
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/// Convergence tolerance
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pub tol: f64,
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/// Depth weighting exponent (0 = none, typically 0.5-0.9)
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pub depth: f64,
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/// Whether sources have free orientation
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pub free_orientation: bool,
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}
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impl Default for EloretaConfig {
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fn default() -> Self {
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Self {
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lambda2: 1.0 / 9.0, // SNR^2 = 9
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max_iter: 100,
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tol: 1e-6,
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depth: 0.5,
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free_orientation: true,
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}
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}
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}
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impl EloretaConfig {
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/// Create config for fixed orientation sources
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pub fn fixed() -> Self {
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Self {
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free_orientation: false,
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..Self::default()
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}
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}
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/// Set regularization parameter
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pub fn with_lambda2(mut self, lambda2: f64) -> Self {
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self.lambda2 = lambda2;
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self
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}
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/// Set depth weighting
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pub fn with_depth(mut self, depth: f64) -> Self {
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self.depth = depth;
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self
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}
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}
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/// eLORETA inverse operator
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#[derive(Debug, Clone)]
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pub struct EloretaInverse {
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/// Inverse kernel [n_source_columns x n_channels]
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kernel: DMatrix<f64>,
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/// Weight matrix for each source (for normalization)
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weights: Vec<DMatrix<f64>>,
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/// Number of sources
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n_sources: usize,
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/// Number of channels
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n_channels: usize,
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/// Whether sources have free orientation
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free_orientation: bool,
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/// Configuration used
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config: EloretaConfig,
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/// Source indices
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source_indices: Vec<usize>,
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}
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impl EloretaInverse {
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/// Create an eLORETA inverse operator
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///
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/// # Arguments
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/// * `gain` - Forward model gain matrix [n_channels x n_sources]
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/// * `noise_cov` - Noise covariance matrix
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/// * `config` - eLORETA configuration
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pub fn make_inverse(
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gain: &GainMatrix,
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noise_cov: &Covariance,
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config: EloretaConfig,
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) -> InverseResult<Self> {
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let n_channels = gain.n_sensors();
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let n_source_cols = gain.n_source_columns();
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let free_orientation = gain.is_free_orientation();
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if noise_cov.n_channels() != n_channels {
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return Err(InverseError::DimensionMismatch(format!(
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"Noise covariance has {} channels, gain has {}",
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noise_cov.n_channels(),
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n_channels
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)));
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}
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// Convert gain to nalgebra matrix
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let g = Self::gain_to_matrix(gain);
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// Compute inverse of regularized noise covariance
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let c_inv = Self::compute_regularized_inv(noise_cov, config.lambda2)?;
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// Compute depth weights
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let depth_weights = Self::compute_depth_weights(&g, config.depth, free_orientation);
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// Compute eLORETA weights iteratively
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let (weights, kernel) = if free_orientation {
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Self::compute_eloreta_weights_free(&g, &c_inv, &depth_weights, &config)?
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} else {
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Self::compute_eloreta_weights_fixed(&g, &c_inv, &depth_weights, &config)?
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};
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let n_sources = if free_orientation {
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n_source_cols / 3
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} else {
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n_source_cols
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};
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Ok(Self {
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kernel,
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weights,
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n_sources,
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n_channels,
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free_orientation,
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config,
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source_indices: (0..n_sources).collect(),
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})
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}
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/// Apply the inverse operator to sensor data
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///
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/// # Arguments
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/// * `data` - Sensor data [n_channels x n_times]
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///
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/// # Returns
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/// Source estimates
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pub fn apply(&self, data: &[Vec<f64>]) -> InverseResult<SourceEstimate> {
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if data.len() != self.n_channels {
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return Err(InverseError::DimensionMismatch(format!(
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"Expected {} channels, got {}",
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self.n_channels,
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data.len()
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)));
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}
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let n_times = data[0].len();
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// Convert to matrix
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let data_mat = DMatrix::from_fn(self.n_channels, n_times, |i, j| data[i][j]);
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// Apply inverse kernel
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let source_mat = &self.kernel * &data_mat;
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// Convert to output format
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let n_rows = source_mat.nrows();
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let source_data: Vec<Vec<f64>> = (0..n_rows)
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.map(|i| (0..n_times).map(|j| source_mat[(i, j)]).collect())
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.collect();
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let times: Vec<f64> = (0..n_times).map(|i| i as f64).collect();
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Ok(SourceEstimate::new(
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source_data,
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times,
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self.source_indices.clone(),
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self.free_orientation,
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))
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}
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/// Get the inverse kernel matrix
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pub fn kernel(&self) -> &DMatrix<f64> {
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&self.kernel
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}
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/// Get number of sources
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pub fn n_sources(&self) -> usize {
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self.n_sources
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}
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/// Get the weight matrices
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pub fn weights(&self) -> &[DMatrix<f64>] {
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&self.weights
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}
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// ========== Private methods ==========
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/// Convert GainMatrix to nalgebra DMatrix
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fn gain_to_matrix(gain: &GainMatrix) -> DMatrix<f64> {
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let n_rows = gain.n_sensors();
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let n_cols = gain.n_source_columns();
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let data = gain.data();
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DMatrix::from_fn(n_rows, n_cols, |i, j| data[i][j])
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}
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/// Compute regularized inverse of noise covariance
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fn compute_regularized_inv(
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noise_cov: &Covariance,
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lambda2: f64,
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) -> InverseResult<DMatrix<f64>> {
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let n = noise_cov.n_channels();
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let c = noise_cov.data();
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// Add regularization: C_reg = C + lambda^2 * trace(C)/n * I
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let trace: f64 = (0..n).map(|i| c[(i, i)]).sum();
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let reg_val = lambda2 * trace / n as f64;
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let mut c_reg = c.clone();
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for i in 0..n {
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c_reg[(i, i)] += reg_val;
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}
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// Compute inverse via eigendecomposition for stability
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let eigen = c_reg.symmetric_eigen();
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let eigenvalues = eigen.eigenvalues;
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let eigenvectors = eigen.eigenvectors;
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let min_eig = eigenvalues.iter().copied().fold(f64::INFINITY, f64::min);
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if min_eig <= 0.0 {
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return Err(InverseError::ComputationError(format!(
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"Regularized covariance has non-positive eigenvalue: {:.2e}",
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min_eig
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)));
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}
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let d_inv = DMatrix::from_diagonal(&DVector::from_fn(n, |i, _| 1.0 / eigenvalues[i]));
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Ok(&eigenvectors * &d_inv * eigenvectors.transpose())
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}
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/// Compute depth weights
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fn compute_depth_weights(gain: &DMatrix<f64>, depth: f64, free_orientation: bool) -> Vec<f64> {
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let n_cols = gain.ncols();
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if depth == 0.0 {
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return vec![1.0; n_cols];
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}
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if free_orientation {
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let n_sources = n_cols / 3;
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let mut weights = Vec::with_capacity(n_cols);
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for src in 0..n_sources {
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// Compute norm across all orientations
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let mut sum_sq = 0.0;
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for ori in 0..3 {
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let col_idx = 3 * src + ori;
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for row in 0..gain.nrows() {
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sum_sq += gain[(row, col_idx)].powi(2);
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}
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}
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let norm = sum_sq.sqrt();
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let w = if norm > 1e-15 { norm.powf(-depth) } else { 1.0 };
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// Same weight for all orientations
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weights.push(w);
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weights.push(w);
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weights.push(w);
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}
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weights
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} else {
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(0..n_cols)
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.map(|col| {
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let sum_sq: f64 = (0..gain.nrows()).map(|row| gain[(row, col)].powi(2)).sum();
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let norm = sum_sq.sqrt();
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if norm > 1e-15 { norm.powf(-depth) } else { 1.0 }
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})
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.collect()
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}
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}
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/// Compute eLORETA weights for fixed orientation sources
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fn compute_eloreta_weights_fixed(
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gain: &DMatrix<f64>,
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c_inv: &DMatrix<f64>,
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depth_weights: &[f64],
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config: &EloretaConfig,
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) -> InverseResult<(Vec<DMatrix<f64>>, DMatrix<f64>)> {
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let n_channels = gain.nrows();
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let n_sources = gain.ncols();
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// Initialize weights to identity (scaled by depth)
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let mut source_weights: Vec<f64> = depth_weights.to_vec();
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// Iterative weight computation
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for _iter in 0..config.max_iter {
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let old_weights = source_weights.clone();
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// Compute H = sum_i (w_i^2 * g_i * g_i^T)
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let mut h = DMatrix::zeros(n_channels, n_channels);
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for i in 0..n_sources {
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let g_i = gain.column(i);
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let w_sq = source_weights[i].powi(2);
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for row in 0..n_channels {
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for col in 0..n_channels {
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h[(row, col)] += w_sq * g_i[row] * g_i[col];
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}
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}
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}
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// Regularize H
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let trace: f64 = (0..n_channels).map(|i| h[(i, i)]).sum();
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let reg = config.lambda2 * trace / n_channels as f64;
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for i in 0..n_channels {
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h[(i, i)] += reg;
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}
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// Compute T = (H + λC)^(-1)
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let h_reg = &h + c_inv * config.lambda2;
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let t = match h_reg.clone().try_inverse() {
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Some(inv) => inv,
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None => {
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return Err(InverseError::ComputationError(
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"Failed to invert H matrix".to_string(),
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));
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}
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};
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// Update weights: w_i = (g_i^T * T * g_i)^(-1/4)
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for i in 0..n_sources {
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let g_i = gain.column(i);
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let tg = &t * g_i;
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let gtg = g_i.dot(&tg);
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if gtg > 1e-30 {
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source_weights[i] = depth_weights[i] * gtg.powf(-0.25);
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}
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}
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// Check convergence
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let change: f64 = source_weights
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.iter()
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.zip(old_weights.iter())
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.map(|(a, b)| (a - b).abs())
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.sum::<f64>()
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/ n_sources as f64;
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if change < config.tol {
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break;
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}
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}
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// Compute final kernel: K = W * G^T * T
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// where W is diagonal weight matrix
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let mut h = DMatrix::zeros(n_channels, n_channels);
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for i in 0..n_sources {
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let g_i = gain.column(i);
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let w_sq = source_weights[i].powi(2);
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for row in 0..n_channels {
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for col in 0..n_channels {
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h[(row, col)] += w_sq * g_i[row] * g_i[col];
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}
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}
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}
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let trace: f64 = (0..n_channels).map(|i| h[(i, i)]).sum();
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let reg = config.lambda2 * trace / n_channels as f64;
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for i in 0..n_channels {
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h[(i, i)] += reg;
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}
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let h_reg = &h + c_inv * config.lambda2;
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let t = h_reg.try_inverse().ok_or_else(|| {
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InverseError::ComputationError("Failed to compute final inverse".to_string())
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})?;
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// Kernel [n_sources x n_channels]
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let mut kernel = DMatrix::zeros(n_sources, n_channels);
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for i in 0..n_sources {
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let g_i = gain.column(i);
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let w_sq = source_weights[i].powi(2);
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let tg = &t * g_i;
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for ch in 0..n_channels {
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kernel[(i, ch)] = w_sq * tg[ch];
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}
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}
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// Convert weights to matrix format
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let weight_matrices: Vec<DMatrix<f64>> = source_weights
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.iter()
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.map(|&w| DMatrix::from_element(1, 1, w))
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.collect();
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Ok((weight_matrices, kernel))
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}
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/// Compute eLORETA weights for free orientation sources
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fn compute_eloreta_weights_free(
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gain: &DMatrix<f64>,
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c_inv: &DMatrix<f64>,
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depth_weights: &[f64],
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config: &EloretaConfig,
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) -> InverseResult<(Vec<DMatrix<f64>>, DMatrix<f64>)> {
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let n_channels = gain.nrows();
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let n_sources = gain.ncols() / 3;
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// Initialize weight matrices (3x3 for each source)
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let mut source_weights: Vec<Matrix3<f64>> = (0..n_sources)
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.map(|i| {
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let d = depth_weights[3 * i];
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Matrix3::identity() * d
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})
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.collect();
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// Iterative weight computation
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for _iter in 0..config.max_iter {
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let old_weights: Vec<Matrix3<f64>> = source_weights.clone();
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// Compute H = sum_i (G_i * W_i^2 * G_i^T)
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let mut h = DMatrix::zeros(n_channels, n_channels);
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for i in 0..n_sources {
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let g_i = Self::extract_source_gain(gain, i);
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let w_sq = source_weights[i] * source_weights[i];
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// H += G_i * W_i^2 * G_i^T
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for ch1 in 0..n_channels {
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for ch2 in 0..n_channels {
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let mut sum = 0.0;
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for ori1 in 0..3 {
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for ori2 in 0..3 {
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sum += g_i[(ch1, ori1)] * w_sq[(ori1, ori2)] * g_i[(ch2, ori2)];
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}
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}
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h[(ch1, ch2)] += sum;
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}
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}
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}
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// Regularize H
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let trace: f64 = (0..n_channels).map(|i| h[(i, i)]).sum();
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let reg = config.lambda2 * trace / n_channels as f64;
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for i in 0..n_channels {
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h[(i, i)] += reg;
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}
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// Compute T = (H + λC)^(-1)
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let h_reg = &h + c_inv * config.lambda2;
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let t = match h_reg.clone().try_inverse() {
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Some(inv) => inv,
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None => {
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return Err(InverseError::ComputationError(
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"Failed to invert H matrix".to_string(),
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));
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}
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};
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// Update weights: W_i = D_i * (G_i^T * T * G_i)^(-1/4)
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for i in 0..n_sources {
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let g_i = Self::extract_source_gain(gain, i);
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|
||||
// Compute G_i^T * T * G_i (3x3 matrix)
|
||||
let mut gtg = Matrix3::zeros();
|
||||
for ori1 in 0..3 {
|
||||
for ori2 in 0..3 {
|
||||
let mut sum = 0.0;
|
||||
for ch1 in 0..n_channels {
|
||||
for ch2 in 0..n_channels {
|
||||
sum += g_i[(ch1, ori1)] * t[(ch1, ch2)] * g_i[(ch2, ori2)];
|
||||
}
|
||||
}
|
||||
gtg[(ori1, ori2)] = sum;
|
||||
}
|
||||
}
|
||||
|
||||
// Compute matrix power -1/4 via eigendecomposition
|
||||
if let Some(new_w) = Self::matrix_power_quarter_inv(>g) {
|
||||
let d = depth_weights[3 * i];
|
||||
source_weights[i] = new_w * d;
|
||||
}
|
||||
}
|
||||
|
||||
// Check convergence
|
||||
let change: f64 = source_weights
|
||||
.iter()
|
||||
.zip(old_weights.iter())
|
||||
.map(|(a, b)| (a - b).norm())
|
||||
.sum::<f64>()
|
||||
/ n_sources as f64;
|
||||
|
||||
if change < config.tol {
|
||||
break;
|
||||
}
|
||||
}
|
||||
|
||||
// Compute final kernel
|
||||
let mut h = DMatrix::zeros(n_channels, n_channels);
|
||||
for i in 0..n_sources {
|
||||
let g_i = Self::extract_source_gain(gain, i);
|
||||
let w_sq = source_weights[i] * source_weights[i];
|
||||
|
||||
for ch1 in 0..n_channels {
|
||||
for ch2 in 0..n_channels {
|
||||
let mut sum = 0.0;
|
||||
for ori1 in 0..3 {
|
||||
for ori2 in 0..3 {
|
||||
sum += g_i[(ch1, ori1)] * w_sq[(ori1, ori2)] * g_i[(ch2, ori2)];
|
||||
}
|
||||
}
|
||||
h[(ch1, ch2)] += sum;
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
let trace: f64 = (0..n_channels).map(|i| h[(i, i)]).sum();
|
||||
let reg = config.lambda2 * trace / n_channels as f64;
|
||||
for i in 0..n_channels {
|
||||
h[(i, i)] += reg;
|
||||
}
|
||||
|
||||
let h_reg = &h + c_inv * config.lambda2;
|
||||
let t = h_reg.try_inverse().ok_or_else(|| {
|
||||
InverseError::ComputationError("Failed to compute final inverse".to_string())
|
||||
})?;
|
||||
|
||||
// Kernel [3*n_sources x n_channels]
|
||||
let mut kernel = DMatrix::zeros(3 * n_sources, n_channels);
|
||||
for i in 0..n_sources {
|
||||
let g_i = Self::extract_source_gain(gain, i);
|
||||
let w_sq = source_weights[i] * source_weights[i];
|
||||
|
||||
for ori in 0..3 {
|
||||
for ch in 0..n_channels {
|
||||
let mut sum = 0.0;
|
||||
for ori2 in 0..3 {
|
||||
for ch2 in 0..n_channels {
|
||||
sum += w_sq[(ori, ori2)] * g_i[(ch2, ori2)] * t[(ch2, ch)];
|
||||
}
|
||||
}
|
||||
kernel[(3 * i + ori, ch)] = sum;
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
// Convert to output format
|
||||
let weight_matrices: Vec<DMatrix<f64>> = source_weights
|
||||
.iter()
|
||||
.map(|w| {
|
||||
let mut m = DMatrix::zeros(3, 3);
|
||||
for i in 0..3 {
|
||||
for j in 0..3 {
|
||||
m[(i, j)] = w[(i, j)];
|
||||
}
|
||||
}
|
||||
m
|
||||
})
|
||||
.collect();
|
||||
|
||||
Ok((weight_matrices, kernel))
|
||||
}
|
||||
|
||||
/// Extract 3-column gain matrix for a single source
|
||||
fn extract_source_gain(gain: &DMatrix<f64>, source_idx: usize) -> DMatrix<f64> {
|
||||
let n_channels = gain.nrows();
|
||||
let start_col = 3 * source_idx;
|
||||
|
||||
DMatrix::from_fn(n_channels, 3, |row, col| gain[(row, start_col + col)])
|
||||
}
|
||||
|
||||
/// Compute A^(-1/4) for a 3x3 symmetric positive definite matrix
|
||||
fn matrix_power_quarter_inv(a: &Matrix3<f64>) -> Option<Matrix3<f64>> {
|
||||
// Convert to nalgebra for eigendecomposition
|
||||
let eigen = a.symmetric_eigen();
|
||||
|
||||
// Check all eigenvalues are positive
|
||||
for &ev in eigen.eigenvalues.iter() {
|
||||
if ev <= 1e-30 {
|
||||
return None;
|
||||
}
|
||||
}
|
||||
|
||||
// Compute D^(-1/4)
|
||||
let d_inv_quarter = Matrix3::from_diagonal(&nalgebra::Vector3::new(
|
||||
eigen.eigenvalues[0].powf(-0.25),
|
||||
eigen.eigenvalues[1].powf(-0.25),
|
||||
eigen.eigenvalues[2].powf(-0.25),
|
||||
));
|
||||
|
||||
// A^(-1/4) = V * D^(-1/4) * V^T
|
||||
Some(eigen.eigenvectors * d_inv_quarter * eigen.eigenvectors.transpose())
|
||||
}
|
||||
}
|
||||
|
||||
#[cfg(test)]
|
||||
mod tests {
|
||||
use super::*;
|
||||
use crate::covariance::CovarianceType;
|
||||
|
||||
fn create_simple_gain() -> GainMatrix {
|
||||
// 4 sensors, 2 sources with free orientation (6 columns)
|
||||
let data = vec![
|
||||
vec![1.0, 0.2, 0.1, 0.3, 0.5, 0.2],
|
||||
vec![0.2, 1.0, 0.3, 0.5, 0.3, 0.1],
|
||||
vec![0.1, 0.3, 1.0, 0.2, 0.1, 0.5],
|
||||
vec![0.3, 0.1, 0.2, 1.0, 0.2, 0.3],
|
||||
];
|
||||
let names = vec![
|
||||
"S1".to_string(),
|
||||
"S2".to_string(),
|
||||
"S3".to_string(),
|
||||
"S4".to_string(),
|
||||
];
|
||||
GainMatrix::new(data, true, names).unwrap()
|
||||
}
|
||||
|
||||
fn create_fixed_gain() -> GainMatrix {
|
||||
// 4 sensors, 3 fixed sources
|
||||
let data = vec![
|
||||
vec![1.0, 0.5, 0.2],
|
||||
vec![0.5, 1.0, 0.5],
|
||||
vec![0.2, 0.5, 1.0],
|
||||
vec![0.1, 0.3, 0.6],
|
||||
];
|
||||
let names = vec![
|
||||
"S1".to_string(),
|
||||
"S2".to_string(),
|
||||
"S3".to_string(),
|
||||
"S4".to_string(),
|
||||
];
|
||||
GainMatrix::new(data, false, names).unwrap()
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_eloreta_config() {
|
||||
let config = EloretaConfig::default();
|
||||
assert!(config.free_orientation);
|
||||
assert!((config.lambda2 - 1.0 / 9.0).abs() < 1e-10);
|
||||
|
||||
let config = EloretaConfig::fixed().with_lambda2(0.1);
|
||||
assert!(!config.free_orientation);
|
||||
assert!((config.lambda2 - 0.1).abs() < 1e-10);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_make_eloreta_fixed() {
|
||||
let gain = create_fixed_gain();
|
||||
let noise_cov = Covariance::identity(4, CovarianceType::Noise);
|
||||
|
||||
let config = EloretaConfig::fixed();
|
||||
let inv = EloretaInverse::make_inverse(&gain, &noise_cov, config).unwrap();
|
||||
|
||||
assert_eq!(inv.n_sources(), 3);
|
||||
assert!(!inv.free_orientation);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_make_eloreta_free() {
|
||||
let gain = create_simple_gain();
|
||||
let noise_cov = Covariance::identity(4, CovarianceType::Noise);
|
||||
|
||||
let config = EloretaConfig::default();
|
||||
let inv = EloretaInverse::make_inverse(&gain, &noise_cov, config).unwrap();
|
||||
|
||||
assert_eq!(inv.n_sources(), 2); // 6 columns / 3 orientations
|
||||
assert!(inv.free_orientation);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_apply_eloreta() {
|
||||
let gain = create_fixed_gain();
|
||||
let noise_cov = Covariance::identity(4, CovarianceType::Noise);
|
||||
|
||||
let config = EloretaConfig::fixed();
|
||||
let inv = EloretaInverse::make_inverse(&gain, &noise_cov, config).unwrap();
|
||||
|
||||
let data = vec![
|
||||
vec![1.0, 0.0],
|
||||
vec![0.0, 1.0],
|
||||
vec![0.5, 0.5],
|
||||
vec![0.2, 0.8],
|
||||
];
|
||||
|
||||
let stc = inv.apply(&data).unwrap();
|
||||
assert_eq!(stc.n_sources(), 3);
|
||||
assert_eq!(stc.n_times(), 2);
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn test_eloreta_kernel_shape() {
|
||||
let gain = create_simple_gain();
|
||||
let noise_cov = Covariance::identity(4, CovarianceType::Noise);
|
||||
|
||||
let inv =
|
||||
EloretaInverse::make_inverse(&gain, &noise_cov, EloretaConfig::default()).unwrap();
|
||||
|
||||
let kernel = inv.kernel();
|
||||
assert_eq!(kernel.nrows(), 6); // 3 orientations * 2 sources
|
||||
assert_eq!(kernel.ncols(), 4); // 4 channels
|
||||
}
|
||||
}
|
||||
Reference in New Issue
Block a user