Function mcpc3ds_combine 

pub fn mcpc3ds_combine<P: AsRef<[f64]>, M: AsRef<[f64]>>(
    phases: &[Vec<P>],
    mags: &[Vec<M>],
    tes: &[f64],
    sigma: [f64; 3],
    echoes: [usize; 2],
    unwrap_method: UnwrapMethod,
    grid: &Grid,
) -> Result<CoilCombinationResult, InputError>

MCPC-3D-S multi-coil phase combination (Eckstein et al., MRM 2018), as in MriResearchTools.jl’s mcpc3ds for 5D (multi-echo, uncombined) input.

Each receive coil carries its own TE-independent phase offset, so uncombined channels cannot simply be summed. The algorithm estimates the offsets from the coil-summed Hermitian inner product (HIP) of two echoes and combines the channels coherently:

1. HIP = Σ_c |S_{1,c}| |S_{2,c}| exp(i(φ_{2,c} − φ_{1,c})) — the coil offsets cancel in the inter-echo phase difference, so the HIP phase is pure field evolution over ΔTE.
2. The HIP phase is unwrapped once (weighted by √|HIP|, on a robust mask of that weight).
3. Per coil, po_c = φ_{1,c} − (TE₁/ΔTE)·unwrapped_HIP, smoothed in the complex domain with the same masked box-filter Gaussian approximation as MriResearchTools (nan_box_smooth_3d_phase; sigma in voxels, mcpc3ds’s default is [10, 10, 5]; the pipeline passes the ROMEO command line’s 7 mm, see crate::pipeline::FieldMappingConfig::phase_offset_sigma_mm).
4. Per echo, S_e = Σ_c |S_{e,c}|² exp(i(φ_{e,c} − po_c)); output phase arg(S_e) and magnitude sqrt(|S_e|).

With a single coil this reduces to phase_offset_removal (with the robust HIP mask).

Arguments

• phases - Wrapped phase, coil-major: phases[coil][echo] (each nx*ny*nz)
• mags - Magnitude, same layout as phases
• tes - Echo times (any consistent unit; only the ratio TE₁/ΔTE is used)
• sigma - Gaussian smoothing sigma in voxels [x, y, z] for the phase offsets
• echoes - Which two echoes form the HIP, default [0, 1]
• unwrap_method - How to unwrap the HIP phase (ROMEO or Laplacian)
• grid - Volume grid

Errors

If there are no coils, fewer than two echoes (the HIP needs two; a single-echo uncombined acquisition cannot be combined this way), the coils do not all have the same number of echoes, magnitude and phase layouts differ, tes does not have one time per echo, the two echoes are the same or out of range, their echo times are equal, or a volume does not match grid.