Module alignment 

Elastic alignment and SRSF (Square-Root Slope Function) transforms.

This module provides phase-amplitude separation for functional data via the elastic framework. Key capabilities:

• srsf_transform / srsf_inverse — SRSF representation and reconstruction
• elastic_align_pair — Pairwise curve alignment via dynamic programming
• elastic_distance — Elastic (Fisher-Rao) distance between curves
• align_to_target — Align a set of curves to a common target
• karcher_mean — Karcher (Fréchet) mean in the elastic metric
• elastic_self_distance_matrix / elastic_cross_distance_matrix — Distance matrices
• reparameterize_curve / compose_warps — Warping utilities

Structs

AlignmentQuality

Comprehensive alignment quality assessment.

AlignmentResult

Result of aligning one curve to another.

AlignmentResultNd

Result of aligning multidimensional (R^d) curves.

AlignmentSetResult

Result of aligning a set of curves to a common target.

ConstrainedAlignmentResult

Result of landmark-constrained elastic alignment.

DecompositionResult

Result of elastic phase-amplitude decomposition.

KarcherMeanResult

Result of the Karcher mean computation.

TsrvfResult

Result of the TSRVF transform.

Enums

TransportMethod

Method for transporting tangent vectors on the Hilbert sphere.

Functions

align_to_target

Align all curves in data to a single target curve.

alignment_quality

Compute comprehensive alignment quality metrics.

amplitude_distance

Compute the amplitude distance between two curves (= elastic distance after alignment).

amplitude_self_distance_matrix

Compute the symmetric amplitude distance matrix (= elastic self distance matrix).

compose_warps

Compose two warping functions: (γ₁ ∘ γ₂)(t) = γ₁(γ₂(t)).

elastic_align_pair

Align curve f2 to curve f1 using the elastic framework.

elastic_align_pair_constrained

Align f2 to f1 with landmark constraints.

elastic_align_pair_nd

Align an R^d curve f2 to f1 using the elastic framework.

elastic_align_pair_with_landmarks

Align f2 to f1 with automatic landmark detection and elastic constraints.

elastic_cross_distance_matrix

Compute the elastic distance matrix between two sets of curves.

elastic_decomposition

Perform elastic phase-amplitude decomposition of two curves.

elastic_distance

Compute the elastic distance between two curves.

elastic_distance_nd

Elastic distance between two R^d curves.

elastic_self_distance_matrix

Compute the symmetric elastic distance matrix for a set of curves.

karcher_mean

Compute the Karcher (Frechet) mean in the elastic metric.

pairwise_consistency

Measure pairwise alignment consistency via triplet checks.

phase_distance_pair

Compute the phase distance between two curves (geodesic distance of optimal warp from identity).

phase_self_distance_matrix

Compute the symmetric phase distance matrix for a set of curves.

reparameterize_curve

Reparameterize a curve by a warping function.

srsf_inverse

Reconstruct a curve from its SRSF representation.

srsf_inverse_nd

Reconstruct an R^d curve from its SRSF.

srsf_transform

Compute the Square-Root Slope Function (SRSF) transform.

srsf_transform_nd

Compute the SRSF transform for multidimensional (R^d) curves.

tsrvf_from_alignment

Compute TSRVF from a pre-computed Karcher mean alignment.

tsrvf_from_alignment_with_method

Compute TSRVF from a pre-computed Karcher mean with configurable transport.

tsrvf_inverse

Reconstruct aligned curves from TSRVF tangent vectors.

tsrvf_transform

Full TSRVF pipeline: compute Karcher mean, then transport SRSFs to tangent space.

tsrvf_transform_with_method

Full TSRVF pipeline with configurable transport method.

warp_complexity

Compute warp complexity: geodesic distance from a warp to the identity.

warp_smoothness

Compute warp smoothness (bending energy): ∫(γ’’)² dt.