Square-Root Velocity (SRV) Representations

Updated 6 October 2025

Square-Root Velocity Representations are a powerful mathematical tool that linearizes curve analysis by mapping curves to normalized velocity fields.
The framework extends to singular and manifold-valued curves using measure-theoretic relaxations, parallel transport, and Lie group actions.
SRV-based algorithms provide deterministic alignment and efficient statistical analysis for applications in bioinformatics, radar processing, and shape statistics.

The square-root velocity (SRV) representation is a transformative mathematical approach for analyzing and comparing curves, primarily in the context of elastic shape analysis. By mapping a curve to its normalized velocity field, the SRV framework linearizes inherently nonlinear geometric problems, enabling the deployment of efficient optimization and statistical techniques within a Hilbert space. This representation is foundational for both theoretical paper and practical algorithms involving curve registration, matching, and shape statistics. Extensions of the SRV methodology address increasingly general spaces of curves—including absolutely continuous, bounded variation, and manifold-valued curves—and have led to innovations in computational geometry, bioinformatics, and machine learning.

1. Mathematical Foundation of SRV Representations

The classical SRV transform for a curve $c:[0,1] \to \mathbb{R}^d$ is defined by

$R(c)(t) = \frac{c'(t)}{\sqrt{|c'(t)|}}, \quad R(c)(t) = 0 \text{ if } c'(t)=0.$

This map projects curves onto a unit sphere in $L^2([0,1],\mathbb{R}^d)$ up to translation, reducing comparisons to standard Hilbert space operations. The induced metric between two curves $c_1, c_2$ is the $L^2$ norm of the difference of their SRV representations: $d(c_1, c_2) = \|R(c_1) - R(c_2)\|_{L^2}.$ Crucially, this metric is reparametrization-invariant after taking the quotient over reparametrizations, i.e., the shape distance between unparametrized curves is

$d_{\mathcal{S}}([c_1],[c_2]) = \inf_{\gamma \in \Gamma} \|R(c_1) - R(c_2 \circ \gamma)\|_{L^2},$

where $\Gamma$ is the (semi)group of admissible reparametrizations $([0,1] \to [0,1])$ —absolutely continuous with boundary and monotonicity constraints (Bruveris, 2015, Grasmair, 2022).

2. SRV Framework: Extensions to Singular and Manifold-Valued Curves

Subsequent research extends the SRV transform from smooth or absolutely continuous curves to broader categories:

Absolutely Continuous Curves: The SRV map becomes a homeomorphism from $AC_0([0,1],\mathbb{R}^d)$ to $L^2([0,1],\mathbb{R}^d)$ , with continuity of the reparametrization action established. Optimal reparametrizations exist for $C^1$ curves but may fail for less regular (e.g., merely Lipschitz) curves (Bruveris, 2015).
Bounded Variation (BV) Curves: For curves where the derivative is a measure (often discontinuous), the SRV distance is defined by relaxing the SRV inner product through measure-theoretic integration, as

$\tilde{S}(c_1, c_2) = \int_{[0,1]} \left( \frac{d|Dc_1|}{d(|Dc_1|+|Dc_2|)} \cdot \frac{d|Dc_2|}{d(|Dc_1|+|Dc_2|)} \right)^+ d(|Dc_1|+|Dc_2|),$

yielding a robust metric $d(c_1, c_2) = \text{len}(c_1) + \text{len}(c_2) - 2\tilde{S}(c_1, c_2)$ (Grasmair, 2022).

Manifold-Valued Curves: For curves $c:[0,1] \to M$ with $M$ a (strong) Riemannian manifold, the framework employs parallel transport or Lie group actions to map velocities to a common tangent space prior to normalization. The SRVT for manifold-valued curves takes the form $R(c)(t) = pt_*c'(t)/\sqrt{|c'(t)|}$ , with $pt_*$ indicating parallel transport or the Maurer–Cartan form (Schmeding, 2016, Su et al., 2017, Celledoni et al., 2017).

3. Algorithmic Implementations for SRV-based Curve Matching

SRV-based alignment algorithms leverage the step-function (piecewise constant) structure of SRVFs for piecewise linear (PL) curves. The optimal matching between two PL curves is constructed as a path $\gamma(z) = (\gamma_1(z), \gamma_2(z))$ in the parameter grid, using weights $W_{ij} = u_i \cdot v_j$ (with $u_i, v_j$ constant segment SRVFs):

Type I (P-segments): traverse grid blocks with positive weights, allowing the matching path's local slope to vary optimally, subject to compatibility conditions:

$H_{i+1,j}/H_{i,j} = \left(\frac{W_{i+1,j}}{W_{i,j}}\right)^2, \quad H_{i,j+1}/H_{i,j} = \left(\frac{W_{i,j}}{W_{i,j+1}}\right)^2.$

Type II (N-segments): traverse blocks with nonpositive weights, forcing the path to be horizontal or vertical.
The optimal matching is a concatenation of P- and N-segments, where consecutive N-segments are excluded and slope transitions follow optimality rules (Lahiri et al., 2015).

This algorithm distinguishes itself from dynamic programming approximations by providing exact (canonical) solutions, exploiting the density of step functions in $L^2$ .

4. Generalization to Lie Groups, Homogeneous Spaces, and Manifold Statistics

SRV analysis has been extended to curves in Lie groups $G$ and homogeneous spaces $M = G/K$ , with analysis conducted via horizontal lifts and the identification of velocities in the Lie algebra $\mathfrak{g}$ :

The generalized SRVT uses right-logarithmic derivatives $\delta^r(c)(t) = T_eR_{c(t)^{-1}}(c'(t))$ for $c$ in $G$ , and transports velocities accordingly for curves in homogeneous spaces.
Metrics pulled back via the SRVT are reparametrization invariant and enable computation of geodesics, Karcher means, and principal component analysis after quotienting out rigid motions and parameter changes.
This metric framework supports large-scale statistical analysis of shape populations, as exemplified by quantification of hurricane tracks on the sphere $S^2 \simeq SO(3)/SO(2)$ , revealing separation between position and intrinsic shape (Su et al., 2017, Celledoni et al., 2017).

5. Computational and Statistical Applications of SRV Methods

The SRV framework is central to geometric and functional data analysis, radar signal processing, and bioinformatics:

It provides a mathematically justified methodology for curve registration (aligning shapes), clustering, and average computation (Fréchet/Karcher mean).
In radar signal processing, spectral evolution is modeled as a curve in the hyperbolic plane, with SRV-induced metrics capturing both spatial location and geometric deformation—facilitating tasks such as target recognition (Brigant, 2016).
In protein LLMs, SRV analysis of representation spaces reveals detailed layerwise transformation characteristics, with computation of the Fréchet mean and effective dimension elucidating structural encoding regimes (Beshkov et al., 29 Sep 2025).

Recent developments include supervised deep learning frameworks to directly estimate SRV distances between discretized curves, leveraging shape-preserving data augmentation for parameterization- and rotation-invariance. These CNN-based systems outperform dynamic programming on computation time while maintaining high accuracy, producing viable tools for large-scale curve analysis (Hartman et al., 2021).

6. Existence and Structure of Optimal Reparametrizations

Analysis of the quotient space of curves modulo reparametrizations is pivotal for "shape space" considerations:

For $C^1$ curves, optimal reparametrization pairings $(\gamma_1,\gamma_2)$ exist, realising minimal shape distances; for curves of only Lipschitz regularity, non-attainment is possible (Bruveris, 2015).
In the BV setting, "generalized reparametrizations" are constructed to resolve composition subtleties at jump points; the shape distance remains invariant, and minimizers exist under appropriate conditions (Grasmair, 2022).
The structure of optimal matchings on PL curves (step functions) is characterized rigorously via concatenations of segments with well-defined slope rules, enabling deterministic alignment based on the SRVF framework (Lahiri et al., 2015).

7. Impact and Future Directions

The SRV framework has established a unifying analytic and computational paradigm for curve and shape analysis. Its continued extension to manifold-valued data, group actions, and deep learning models expands the scope toward new application domains, such as protein representation spaces and radar signature analysis. Current trends emphasize robust metric constructions for irregular data, efficient algorithms for high-dimensional statistics, and interpretable deep learning architectures guided by SRV invariances. The need for well-posedness in optimization over reparametrizations remains an active area of mathematical research, particularly as the framework is adopted for increasingly general function spaces and data modalities.

The SRV representation thereby forms the backbone of modern elastic shape analysis, underpinning rigorous mathematics, efficient algorithms, and diverse data science applications. Its invariance properties and adaptability to manifold structures establish it as a cornerstone tool for geometric and statistical studies of curves and surfaces.