Module warping 

Warping function utilities and Hilbert sphere geometry.

This module provides operations on warping (reparameterization) functions, including their Hilbert sphere representation via ψ(t) = √γ'(t).

Key capabilities:

• gam_to_psi / psi_to_gam — Convert between warping functions and sphere
• exp_map_sphere / inv_exp_map_sphere — Riemannian exponential / log maps
• normalize_warp / invert_gamma — Warp normalization and inversion
• phase_distance — Geodesic distance from a warp to the identity

Functions

exp_map_sphere

Exponential map on the Hilbert sphere. Moves from psi along tangent vector v.

gam_to_psi

Convert warping function to Hilbert sphere representation: ψ = √γ’.

gam_to_psi_smooth

Convert warping function to smoothed Hilbert sphere representation.

inner_product_l2

L2 inner product: ∫ψ₁·ψ₂ dt via trapezoidal rule.

inv_exp_map_sphere

Inverse exponential (log) map on the Hilbert sphere. Returns tangent vector at mu pointing toward psi.

invert_gamma

Invert a warping function: find γ⁻¹ such that γ⁻¹(γ(t)) = t. gam and time are both on [0,1].

l2_norm_l2

L2 norm: √(∫ψ² dt).

normalize_warp

Ensure γ is a valid warping: monotone non-decreasing, with correct boundary values.

phase_distance

Geodesic distance from a warping function to the identity on the Hilbert sphere.

psi_to_gam

Convert ψ back to warping function: γ = cumtrapz(ψ²), normalized to [0,1].