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.

BayesianAlignConfig

Configuration for Bayesian pairwise alignment.

BayesianAlignmentResult

Result of Bayesian pairwise alignment.

ClosedAlignmentResult

Result of aligning one closed curve to another.

ClosedKarcherMeanResult

Result of computing the Karcher mean for closed curves.

ConstrainedAlignmentResult

Result of landmark-constrained elastic alignment.

DecompositionResult

Result of elastic phase-amplitude decomposition.

Dendrogram

Result of hierarchical clustering (dendrogram).

DiagnosticConfig

Configuration for alignment diagnostics.

ElasticDepthResult

Result of elastic depth computation.

ElasticOutlierConfig

Configuration for elastic outlier detection.

ElasticOutlierResult

Result of elastic outlier detection.

FpnsResult

Result of horizontal Functional Principal Nested Spheres (FPNS) analysis.

GenerativeModelResult

Result of Gaussian generative model sampling.

GeodesicPath

A geodesic path between two 1-D curves in the elastic metric.

GeodesicPathNd

A geodesic path between two N-D curves in the elastic metric.

KMedoidsConfig

Configuration for k-medoids clustering.

KMedoidsResult

Result of k-medoids clustering.

KarcherMeanResult

Result of the Karcher mean computation.

KarcherMeanResultNd

Result of the Karcher mean computation for multidimensional (R^d) curves.

LambdaCvConfig

Configuration for lambda cross-validation.

LambdaCvResult

Result of lambda cross-validation.

MultiresConfig

Configuration for multi-resolution alignment.

OrbitRepresentative

A canonical representative of a shape orbit.

PartialMatchConfig

Configuration for elastic partial matching.

PartialMatchResult

Result of elastic partial matching.

PcaNdResult

Result of PCA on aligned multidimensional (R^d) curves.

PersistenceDiagramResult

Result of the peak persistence analysis across a sweep of lambda values.

PhaseBoxplot

Result of computing a phase (warping) box plot.

RobustKarcherConfig

Configuration for robust Karcher estimation.

RobustKarcherResult

Result of robust Karcher estimation.

ShapeCiConfig

Configuration for shape bootstrap confidence intervals.

ShapeCiResult

Result of shape bootstrap confidence interval computation.

ShapeDistanceResult

Result of computing the elastic shape distance between two curves.

ShapeMeanResult

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

TransferAlignConfig

Configuration for transfer alignment.

TransferAlignResult

Result of transfer alignment.

TsrvfResult

Result of the TSRVF transform.

WarpStatistics

Statistics computed on a set of warping functions.

Enums

Linkage

Linkage method for hierarchical clustering.

ShapeQuotient

Quotient space for shape analysis.

TransportMethod

Method for transporting tangent vectors on the Hilbert sphere.

WarpPenaltyType

Penalty type for alignment regularization.

Traits

AlignmentOutput

Common interface for alignment results, enabling interchangeable alignment methods in downstream analysis (elastic FPCA, regression, etc.).

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).

bayesian_align_pair

Perform Bayesian pairwise alignment of f2 to f1 via pCN MCMC on the Hilbert sphere.

compose_warps

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

curve_geodesic

Compute the geodesic path between two 1-D curves in the elastic metric.

curve_geodesic_nd

Compute the geodesic path between two N-D curves in the elastic metric.

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_closed

Align closed curve f2 to closed curve f1 with rotation search.

elastic_align_pair_constrained

Align f2 to f1 with landmark constraints.

elastic_align_pair_multires

Align curve f2 to f1 using multi-resolution elastic alignment.

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_depth

Compute amplitude + phase decomposed elastic depth.

elastic_distance

Compute the elastic distance between two curves.

elastic_distance_closed

Compute the elastic distance between two closed curves.

elastic_distance_nd

Elastic distance between two R^d curves.

elastic_outlier_detection

Detect outlier curves using elastic distances and the Tukey fence.

elastic_partial_match

Find the best elastic partial match of template within target.

elastic_self_distance_matrix

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

gauss_model

Generate random curves from a fitted Gaussian model on aligned data.

hierarchical_from_distances

Hierarchical agglomerative clustering from a precomputed distance matrix.

horiz_fpns

Perform horizontal Functional Principal Nested Spheres (FPNS) analysis.

invert_warp

Compute the inverse of a warping function.

joint_gauss_model

Generate random curves from a joint Gaussian model preserving amplitude-phase correlation.

karcher_covariance_nd

Compute the cross-dimensional covariance matrix of aligned N-d curves.

karcher_mean

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

karcher_mean_closed

Compute the Karcher (Frechet) mean for closed curves.

karcher_mean_nd

Compute the Karcher (Frechet) mean for multidimensional (R^d) curves.

karcher_median

Compute the Karcher median via the Weiszfeld algorithm on the elastic manifold.

kmedoids_from_distances

K-medoids (PAM-style) clustering from a precomputed distance matrix.

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.

pca_nd

Perform PCA on aligned multidimensional (R^d) curves.

peak_persistence

Analyse the stability of peak count in the Karcher mean across a sweep of alignment penalty values.

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.

robust_karcher_mean

Compute a trimmed Karcher mean by removing the most distant curves.

shape_confidence_interval

Compute bootstrap confidence intervals for the elastic Karcher mean.

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.

transfer_alignment

Align curves from a target population to a source population’s coordinate system.

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.