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

AlignmentDiagnostic

Diagnostic information for a single curve’s alignment.

AlignmentDiagnosticSummary

Summary of diagnostics across all curves.

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.

DiagnosticConfig

Configuration for alignment diagnostics.

ElasticClusterConfig

Configuration for elastic k-means clustering.

ElasticClusterResult

Result of elastic k-means clustering.

ElasticDendrogram

Result of hierarchical elastic clustering (dendrogram).

KarcherMeanResult

Result of the Karcher mean computation.

LambdaCvConfig

Configuration for lambda cross-validation.

LambdaCvResult

Result of lambda cross-validation.

OrbitRepresentative

A canonical representative of a shape orbit.

PhaseBoxplot

Result of computing a phase (warping) box plot.

ShapeDistanceResult

Result of computing the elastic shape distance between two curves.

ShapeMeanResult

Result of computing the shape mean of a set of curves.

TsrvfResult

Result of the TSRVF transform.

WarpStatistics

Statistics computed on a set of warping functions.

Enums

ElasticClusterMethod

Linkage method for elastic clustering.

ShapeQuotient

Quotient space for shape analysis.

TransportMethod

Method for transporting tangent vectors on the Hilbert sphere.

WarpPenaltyType

Penalty type for alignment regularization.

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)).

cut_dendrogram

Cut a dendrogram to produce k clusters.

diagnose_alignment

Diagnose alignment quality for every curve after a Karcher mean computation.

diagnose_pairwise

Diagnose a single pairwise alignment.

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_penalized

Align two curves with a configurable penalty type.

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_hierarchical

Hierarchical elastic clustering using Fisher-Rao distances.

elastic_kmeans

Elastic k-means clustering using Fisher-Rao distances and Karcher means.

elastic_self_distance_matrix

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

invert_warp

Compute the inverse of a warping function.

karcher_mean

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

lambda_cv

Select the best elastic-alignment regularisation parameter via K-fold cross-validation.

orbit_representative

Compute the canonical orbit representative of a curve.

pairwise_consistency

Measure pairwise alignment consistency via triplet checks.

phase_boxplot

Compute a phase box plot from a set of warping functions.

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.

shape_distance

Compute the elastic shape distance between two curves.

shape_mean

Compute the Karcher (Frechet) mean in the elastic shape space.

shape_self_distance_matrix

Compute the pairwise shape distance matrix for a set of curves.

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_inverse_error

Verify roundtrip accuracy: max |γ(γ⁻¹(t)) - t| over the domain.

warp_smoothness

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

warp_statistics

Compute summary statistics for a set of warping functions.