Terminology#

A quick reference for the key concepts used throughout this documentation.

HEALPix#

Key properties:

All cells have the same area (equal-area projection).
Cells are organized in a hierarchical structure: each cell at level l contains exactly 4 sub-cells at level l+1.
Resolution is controlled by level: nside = 2**level, total cell count = 12 * nside².

Nested vs. Ring indexing#

HEALPix cells can be indexed in two schemes:

Nested (nest=True, default): cells with similar indices are spatially close. Best for hierarchical operations.
Ring: cells are indexed by iso-latitude rings. Used in some harmonic analysis tools.

healpix-resample uses nested indexing by default.

Sparse operator M and MT#

The library represents the mapping between sample points and HEALPix cells as two sparse matrices:

M of shape (N, K): maps a HEALPix field (on K kept cells) → values at N sample locations.
MT of shape (K, N): maps sample values (N) → accumulated values on K HEALPix cells.

These operators are built once during initialization and reused across all calls to resample().

N, K, B#

Throughout the documentation:

N — number of input sample points (lon/lat measurements).
K — number of HEALPix cells that receive at least one contribution (a subset of all cells at the given level).
B — batch size (number of fields to process at once, e.g. time steps or variables).

Threshold#

The threshold parameter controls which HEALPix cells are kept. After computing accumulated Gaussian weights from all samples, only cells whose total weight exceeds threshold are retained. This keeps the operator compact and avoids nearly-empty cells polluting results.

Lower threshold → more cells kept, including weakly covered ones.
Higher threshold → only well-supported cells are kept.

Sigma (σ)#

sigma_m is the Gaussian length scale in metres used to compute the weight of each sample-to-cell link:

w = exp(-2 * d² / σ²)

where d is the geodesic distance between the sample and the cell centre. If sigma_m is not set, the library defaults to the pixel size at the chosen level:

σ = sqrt(4π / (12 × 4^level)) × R_Earth

Conjugate Gradient (CG)#

Used by PSFResampler to solve the damped least-squares problem:

minimize ‖ (x_ref + δ) @ MT - y ‖² + λ ‖δ‖²

The CG solver iteratively refines the solution without forming any dense matrix. lam (λ) is the Tikhonov regularization strength — set it higher when your data is sparse or noisy.

ResampleResults#

The dataclass returned by all resample() calls:

Field	Shape	Description
`cell_data`	`(K,)` or `(B, K)`	Values on the HEALPix cells
`cell_ids`	`(K,)`	HEALPix cell indices (nested scheme)
`cg_residual_norms`	`(iters,)`	CG convergence history (PSFResampler only)
`cg_niters`	scalar	Number of CG iterations (PSFResampler only)